ENDICONTI EMINARIO ATEMATICOcalvino.polito.it/~gatto/public/Syzygy2005/64_4.pdfD. Gallarati and V....

RENDICONTI

DEL SEMINARIO

MATEMATICO

Universita e Politecnico di Torino

Syzygy 2005

CONTENTS

P. Salmon, The scientific work of Paolo Valabrega . . . . . . . . . . . . . . . . 343

SURVEY PAPERS

R. Achilles – M. Manaresi, Generalized Samuel multiplicities and applications 345

D.A. Buchsbaum, Alla ricerca delle risoluzioni perdute . . . . . . . . . . . . . 373

L. Chiantini, Vector bundles, reflexive sheaves and low codimensional varieties 381

E. Gover, Maximal Poincare series and bounds for Betti numbers . . . . . . . . 407

R. Hartshorne, Liaison with Cohen–Macaulay modules . . . . . . . . . . . . . 419

S. Nollet, Deformations of space curves: connectedness of Hilbert schemes . . 433

RESEARCH PAPERS

R.M. Miro-Roig – R. Notari – M.L. Spreafico, Properties of some Artinian

Gorenstein rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

N. Mohan Kumar – A.P. Rao – G.V. Ravindra, Four-by-four Pfaffians . . . . . . 471

G. Restuccia, Symmetric algebras of finitely generated graded modules and s-

sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Volume 64, N. 4 2006

DIRETTORE

CATTERINA DAGNINO

COMMISSIONE SCIENTIFICA (2006–08)

C. Dagnino (Direttore), R. Monaco (Vicedirettore), G. Allasia, S. Benenti, A. Collino,

F. Fagnani, G. Grillo, C. Massaza, F. Previale, G. Zampieri

COMITATO DIRETTIVO (2006–08)

S. Console, S. Garbiero, G. Rossi, G. Tedeschi, D. Zambella

Proprieta letteraria riservata

Autorizzazione del Tribunale di Torino N. 2962 del 6.VI.1980

Direttore Responsabile: CATTERINA DAGNINO

QUESTO FASCICOLO E STAMPATO CON IL CONTRIBUTO DI:

UNIVERSITA DEGLI STUDI DI TORINO

POLITECNICO DI TORINO

Preface

The International Italian Conference “Syzygy 2005” on Commutative Algebra

and Algebraic Geometry dedicated to Paolo Valabrega on the occasion of his 60th birth-

day was held on February 18–20, 2005 at the Politecnico of Torino, Italy (although the

Conference was in advance with respect to Valabrega’s birthday, which is May, 29).

This volume contains a presentation of the scientific activity of Paolo Valabrega and

some articles related with the talks given in the meeting written by some of the invited

speakers and their collaborators or members of the Scientific Committee. More pre-

cisely, six articles have an essentially expository character and the three others articles

are refereed research papers.

Speakers: David Buchsbaum (Brandeis University, USA), Anthony Geramita

(Universita di Genova, Italy), Eugene Gover (NorthEastern University, USA), Robin

Hartshorne (University of California, USA), Mirella Manaresi (Universita di Bologna,

Italy), Rosa Maria Miro-Roig (Universidad de Barcelona, Spain), Scott Nollet (Texas

Christian University, USA), A. Prabhakar Rao (University of Missouri, USA), Gaetana

Restuccia (Universita di Messina, Italy), Giuseppe Valla (Universita di Genova, Italy),

Edoardo Vesentini (Politecnico di Torino, Italy).

Scientific Committee: L. Chiantini, M. Roggero, P. Salmon.

Local Organizing Committee: G. Beccari, R. Camerlo, E. Carlini, J. Cordovez,

C. Cumino, R. Di Nardo, A. J. Di Scala, L. Gatto, C. Massaza, L. Motto Ros, J. Pejsa-

chowicz, S. Salamon, T. Santiago, G. Tedeschi, M. Valenzano.

Editors of the Proeedings: L. Chiantini, P. Salmon, M. Valenzano.

A particular high contribution in the organization of the Conference was given

by CarlaMassaza and Letterio Gatto, while the preparation of these proceedings brought

advantage of Mario Valenzano’s work.

A special thank has to be addressed to the pianist Maria Iovino, who played

musics of Scarlatti, Mozart and Beethoven in a concert offered to mathematicians in

the afternoon of February 18, which got a remarkable success.

Last but not least, many thanks to the following institutions for their financial

and/or logistic support to the Conference: Dipartimento di Matematica del Politecnico

di Torino, Scuola di Dottorato del Politecnico di Torino, Dipartimento di Matematica

dell’Universita di Torino, Dipartimento di Matematica dell’Universita di Bologna, Di-

partimento di Matematica dell’Universita di Siena, GNSAGA-INdAM, MIUR national

project “Geometry on Algebraic Varieties”, ISMB Istituto Superiore Mario Boella,

Polincontri Classica, SanPaolo IMI, UniCredit Banca, Poste Italiane.

On the behalf of the Editorial Board of the Proceedings

Paolo Salmon

Rend. Sem. Mat. Univ. Pol. Torino - Vol. 64, 4 (2006)

Syzygy 2005

P. Salmon

THE SCIENTIFIC WORK OF PAOLO VALABREGA

Dedicated to Paolo Valabrega on the occasion of his 60 th birthday

Paolo Valabrega got his degree as “dottore in Scienze Matematiche” in the year 1968,

discussing a thesis on some problems related to general topology under the direction

of D. Demaria. He published soon a first article on a topological subject and, almost

immediately later, a second one of historical interest on Hilbert’s “Grundlagen der

Geometrie” in cooperation with E. Valabrega.

Starting with 1969, the main scientific interest of Paolo V. moved to two sub-

jects, streectly connected with each other: Commutative Algebra and Algebraic Geom-

etry. More precisely, in the period 1969–1980, Paolo V. studied essentially problems

related with Commutative Algebra, while, starting 1980, the main interest became Al-

gebraic Geometry, mainly algebraic curves.

The first contributions of Paolo V. to problems in Commutative Algebra are

related with the scientific activity promoted, first of all in Pisa and then in Genova, as

a consequence of my personal experience in Paris, working there, in the years 1962

and 1963, under the scientific direction of P. Samuel. In that period P. Samuel was

particularly interested in factorial rings, and he could exibhite, among several results,

the first well known examples of non factorial power series rings over a basic factorial

ring. My work in Paris did concern essentially the factoriality of restricted power series

rings over a regular and factorial ring A with respect to a suitablem-adic topology fixedin A. In the following years, Paolo V. too was inerested in restricted power series.

When, in the summer of 1969, Paolo Valabrega came to Genova with the pro-

posal of a collaboration with our group working in Commutative Algebra, among the

people inside that group (living in Genova and outside) one may mention: P. Corsini,

M. Fiorentini, S. Greco, S. Guazzone, T. Millevoi, C. Pedrini, L. Robbiano, P. Salmon,

G. Valla, G. Vecchio. The principal objects in their research in Commutative Algebra

were: factorial rings, algebraic properties of m-adic completions, henselian rings andtheir first generalizations by Lafon and Greco, projective modules, Picard groups, sym-

metric algebras and Rees algebras related to ideals. D. Gallarati and V. Villani, both

professors of Geometry in Genova and interested respectively in Algebraic Geome-

try and Complex Geometry, very often offered their cooperation with the Commuative

Algebra group: some of their students did work on problems of common interest.

The starting point of the first research developed by Paolo V. in the field of

Commutative Algebra was the classical Hensel’s Lemma, stating that if (A,m) is acomplete local ring and f is a unitary polynomial in the polynomial ring A[X ] withimage f ∗ in (A/m)[X ], a decomposition of f ∗ in a product of relatively prime poly-nomials in (A/m)[X ] can be lifted to a decomposition of f in A[X ]. In Chapter 4of Bourbaki’s Algebre Commutative, appeared in 1961, there was a generalization of

343

344 P. Salmon

the above statement involving, beside a unitary polynomial, also restricted power se-

ries rings with respect to linear topologies where the basic commmutative ring A is

not necessarily local and the role of the maximal ideal in the classical statement it was

assumed by a closed ideal m of A.

This generalization suggested to Paolo V. some investigations related with the

new Hensel’s Lemma and he was able to write three articles (appeared around 1972)

concerning properties of henselian topological rings, topological henselizations and

further extentions with applications to valuation rings.

The articles written by Paolo V. in the following years 1972–1980 are primar-

ily concerned with excellent rings, starting with statements of the excellent property

in some topological rings: convergent series and restricted power series rings, com-

pletions of excellent rings (one article was written in collaboration with S. Greco). A

result, particularly appreciated by H. Matsumura, states the excellent property for the

ring of formal power series over a polynomial ring.

In the same period Paolo V. published also some articles on openness of loci

in a particular scheme, lifting properties and special morphisms (one article was in

collaboration with C. Massaza).

In the years 1978–1980 Paolo V. wrote two articles, working together with

G. Valla, on graded rings (form rings and regular sequences, standard bases). Both

articles were studied and highly appreciated by several algebraic experts in the field,

also in recent years.

Starting with 1980, Paolo V. focused his interest on subjects more specifically

related with Algebraic Geometry. In that direction he started a collaboration with

S. Greco on the theory of adjoint curves and, later, on the singularities of algebraic

varieties containing a fixed algebraic curve.

Then, Paolo V.’s interest in Algebraic Geometry had a further and permanent

increasing, with a particular interest for subcanonical curves. He wrote on that sub-

ject several articles, sometimes in collaboration with L. Chiantini, A. Geramita and

M. Roggero.

Going on, the collaboration with M. Roggero, which began before 1990, had

a resolute prominence in the following years and until recent times; it concerned sub-

canonical curves and surfaces, vector bundles, reflexive sheaves and complete intersec-

tions.

Paolo SALMON, Dipartimento di Matematica, Universita di Bologna, Piazza di Porta

S. Donato 5 - 40126 Bologna - ITALIA

e-mail: [email protected]


Syzygy 2005

R. Achilles – M. Manaresi

GENERALIZED SAMUEL MULTIPLICITIES AND

APPLICATIONS


Abstract. In this note we survey and discuss the main results on the multiplicity sequence we

introduced in former papers as a generalization of Samuel’s multiplicity. We relate this new

multiplicity to other numbers introduced in different contexts, for example the Segre numbers

of Gaffney and Gassler and the Hilbert coefficients defined by Ciuperca. Discussing some

examples we underline the usefulness of the multiplicity sequence for concrete calculations

in algebraic geometry using computer algebra systems.

1. Introduction

Intersection theory and singularity theory stimulated the development of a theory of

multiplicities in local rings.

For example, the Samuel multiplicity of the maximal ideal m of a local ring

(A,m) measures the singularity of a variety at some point or along some subvarieties,and the Samuel multiplicity of anm-primary ideal was introduced in order to define theintersection number of an irreducible component of the intersection of two varieties X

and Y .

If X and Y intersect improperly, one must assign intersection numbers to cer-

tain embedded components of X ∩ Y , see for example [18] and [16]. In Fulton and

MacPherson’s approach this is done without a preliminary study of intersection mul-

tiplicity, since their construction gives a well defined cycle, whose coefficients are the

intersection multiplicities, which coincide with Samuel’s multiplicities in the case of

proper intersections. In any case the theory developed by Fulton and MacPherson gives

a motivation to define algebraic multiplicities, which extend Samuel’s one and can pro-

vide intersection numbers.

In Stuckrad and Vogel’s approach (see [16]) some components of the intersec-

tion cycle are defined over the base field k and are called k-rational (see Definition 1,

Section 3.2), and in [2] it was proved that they correspond to ideals of maximal analytic

spread. In [3] it was defined a multiplicity j (I, A) for such ideals I which generalizesSamuel’s multiplicity. Later in [4] this construction was extended to arbitrary ideals in

a local ring. The new algebraic multiplicity for an ideal I of a d-dimensional local ring

(A,m) is a sequence of non-negative integers c(I ) := (c0(I, A), . . . , cd(I, A)) whichcan be used to describe the degrees of the intersection cycle in the sense of [16]. The

first number c0(I, A) coincides with the multiplicity j (I, A) defined in [3].

In this note we survey and discuss the main results on the multiplicity sequence

c(I ) and relate this sequence to other numbers introduced in different contexts, for ex-ample the Segre numbers introduced by T. Gaffney and R. Gassler [19] and the Hilbert

345

346 R. Achilles – M. Manaresi

coefficients defined by C. Ciuperca [10]. Furthermore we give some examples which

show the usefulness of the multiplicity sequence for concrete calculations in algebraic

geometry using computer algebra systems.

Under certain hypothesis the j-multiplicity can be expressed as a length multi-

plicity (see 3.3) and can be computed by an intersection algorithm using generic filter-

regular sequences called “super-reductions”. In 3.4 the precise meaning of “generic”

will be given.

Flenner and Manaresi [14] used the j-multiplicity to give a numerical charac-

terization of reduction ideals, which generalizes a classical theorem of E. Boger (see

3.5). Later C. Ciuperca generalized this result using the j-multiplicity and second

highest Hilbert coefficients, giving a numerical characterization of the S2-closure of

the extended Rees algebra, see Theorem 9.

The multiplicity sequence and the j-multiplicity can be calculated using inter-

section algorithms defined in [3] and in [4], which will be presented in Section 3.6.

These algorithms have been used by several authors, who sometimes produced inter-

esting modifications of them. Here we will refer only to the original versions.

In 1999, T. Gaffney and R. Gassler introduced Segre numbers of an ideal in

a local ring of an analytic space that turned out to be very useful in the study of the

equisingularity of families of hypersurfaces with non-isolated singularities. The Segre

numbers of the Jacobian ideal are simply the generic Le numbers of D. B. Massey [30].

In analytic intersection theory, P. Tworzewski [47] (see also Remark 5) defined the ex-

tended index of intersection, which, in the case of improper intersections, replaces the

classical intersection number of a proper intersection component by a set of numbers.

In 3.7 we will show that all these new invariants can be considered as generalizations

of Samuel’s multiplicity of an m-primary ideal in a noetherian local ring (A,m) to anarbitrary ideal.

In 3.8 we present an application of generalized Samuel multiplicities to singu-

larity theory, more precisely to Whitney stratification of surfaces (see [5]), and discuss

an example.

The paper is divided into two parts: in Section 2 we review some classical re-

sults on Samuel’s multiplicity whose analogues for the generalized Samuel multiplicity

will be presented in Section 3, which is devoted to this new multiplicity, its properties

and its relations to other important invariants.

Notation. In this paper all rings are assumed to be noetherian and the dimension of

a ring means its Krull dimension. A (noetherian) local ring (A,m) is formally equidi-mensional (or, in Nagata’s terminology, quasi-unmixed) if each minimal prime ideal pin the m-adic completion A satisfies dim( A/p) = dim( A). For the properties of for-mally equidimensional local rings we refer to [23], (18.17). In particular we recall that

if A is a formally equidimensional local ring and I is an ideal of A, then the associ-

ated graded ring GI (A) of A with respect to I is formally equidimensional, see[23],(18.24).

We denote the n-dimensional projective space over a field K by PnK and the

Generalized Samuel multiplicities and applications 347

affine space by AnK , respectively. If not explicitly stated the contrary, our base field

will always be algebraically closed and we simply write Pn or An . By a variety or

subvariety of Pn we mean a closed reduced (but possibly reducible) equidimensionalsubscheme of Pn without embedded components. A surface is a 2-dimensional variety,a hypersurface is an (n − 1)-dimensional subvariety. Sometimes, for simplicity, wewill state the results for varieties, but most of them hold for algebraic schemes too.

2. Some classical results on Samuel’s multiplicity

Following [39], in this section we review some classical results on Samuel’s multiplic-

ity whose analogues for the generalized Samuel multiplicity will be presented in the

next section.

2.1. The Samuel multiplicity

Let (A,m) be a d-dimensional local ring and let I be an m-primary ideal.

For each non negative integer j let

H(0)I ( j) := length(I j/I j+1)

and

H(1)I ( j) :=

j∑

k=0

H(0)I ( j) = length(A/I j+1) .

Note that these lengths are finite since I is m-primary. It is well known that for all

sufficiently large j the function H(1)I becomes a polynomial, the so called Hilbert-

Samuel polynomial, which can be written in the form

e0

(j + d

d

)− e1

(j + d − 1d − 1

)+ · · · + (−1)ded ,

where e0, e1, . . . , ed are integers and e0 ≥ 1. The positive integer

e(I, A) := e0

is called Samuel multiplicity of I in A. Sometimes, when the ring is clear from the

context, the multiplicity e(I, A) will be denoted by e(I ). In the case I = m we write

simply

e(A) := e(m, A).

It is immediate that for each couple of m-primary ideals J ⊂ I ⊂ A one has e(J ) ≥e(I ).


2.2. Samuel multiplicity as an intersection number

Samuel’s multilplicity can be applied in the following situation.

Let X, Y be subvarieties ofAn (or Pn). Assume that Y is a complete intersectiondefined by the ideal I (Y ) and let C be an irreducible component of X ∩ Y . Let A :=OX∩Y,C and let I := I (Y )A. Under these assumptions the ideal I is primary withrespect to the maximal ideal of the local ring A and e(I, A) is the intersection numberi(X,Y ;C) of X and Y along C , see [40].

We remark that Samuel proposed this definition of intersection number without

any assumption on the dimension of C . We recall that it is always

dimC ≥ dim X + dim Y − n

and C is called a proper component of X ∩ Y if equality holds, an improper compo-

nent otherwise. Samuel only assumed C to be an irreducible (isolated) component of

X ∩ Y and in [42] Stuckrad and Vogel could prove a theorem of Bezout for improper

intersections counting irreducible components with Samuel’s multiplicities.

2.3. Samuel multiplicity as a length

Let (A,m) be local ring, I ⊂ A an m-primary ideal generated by a system of parame-ters. One has

(1) length(A/I ) ≥ e(I, A)

and equality holds if and only if A is the Cohen–Macaulay (see, for example, [43],

Theorem 1.2 and Lemma 1.3).

D. Buchsbaum conjectured that the difference

length(A/I ) − e(I, A)

was an invariant of A, that is, independent of the choice of I , but it turned out that the

conjecture holds only for the so called Buchsbaum rings, which form a class of local

rings containing the Cohen–Macaulay rings (see [43]).

In Cohen–Macaulay rings the inverse of (1) holds, that is

(2) e(I, A) ≥ length(A/I ) ,

and, if A/m is infinite, equality holds if and only if I can be generated by d = dim A

elements, that is, by a system of parameters (see for example [18], Example 4.3.5.).

We remark that, without the Cohen–Macaulay assumption on A, when the ideal

I is generated by a system of parameters, then one can construct an ideal J containing

I such that

length(A/J ) = e(I, A).


The ideal J is obtained from I by the following intersection algorithm (see [7]). Let

a1, . . . , ad be a system of parameters generating I (observe that an m-primary ideal Ineeds at least d generators) and, for an ideal K of A, let U (K ) denote the intersectionof all primary ideals Q associated to A/K such that dim A/Q = dim A/K . Then set

I0 := (0) and

Ik := ak A +U (Ik−1) for k = 1, . . . , d.

The last ideal Id ⊇ I is the desired ideal J such that

e(I, A) = length(A/Id),

see [7], Proposition 1.

2.4. Reduction ideals and Samuel multiplicity

Let (A,m) be a d-dimensional local ring and let I ⊂ A be an ideal. An ideal J ⊂ I is

called a reduction of I if

J I n = I n+1 for at least one positive integer n.

J is called a minimal reduction of I if no ideal strictly contained in J is a reduction of

I .

Reductions can be described using the integral closure of ideals. Recall that if

I ⊂ A is an ideal, the integral closure I of I is the ideal of A defined by

I = {x ∈ A | ∃ m positive integer and, for i = 1, . . . ,m, elements ai ∈ I i

such that xm + a1xm−1 + · · · + am = 0}.

J ⊆ I is a reduction of I if and only if I ⊆ J (see [32] p. 34 ex. 4 and [28] p. 112).

Denote by G := GI A :=⊕

n≥0 In/I n+1 the graded ring of A with respect to

I . The analytic spread of I can be defined as

s(I ) := dim(G/mG) = dim(G ⊗A k)

and one has

height(I ) ≤ dim A − dim A/I ≤ s(I ) ≤ dim A ,

where height(I ) denotes the height of the ideal I (see [34], p. 151).

The notions of reduction and analytic spread were introduced by Northcott and

Rees in 1954 (see [34]), who proved that if k is infinite then minimal reductions exist

and are minimally generated by s(I ) elements.

If I is m-primary, then s(I ) = dim A and a minimal reduction J of I is gener-

ated by a system of parameters. In this case the algorithm of (2.3) gives

e(J, A) = length(A/Id).


Moreover, if J is a reduction of I , then e(I, A) = e(J, A). In fact, for large m the

lengthA(A/Im) is given by a polynomial of degree d = dim A, which can be written

in the form

hI (m) = e(I )md

d!+ terms of degree < d .

If J I n = I n+1, for each integer m ≥ 1 we have Jm I n = I n+m , hence hI (m + n) ≥hJ (m) ≥ hI (m) which implies e(I ) = e(J ).

EXAMPLE 1. If A = C[[x, y]], I = (x3, y2, x2y), J = (x3, y2), then I J = I 2

and e(I ) = e(J ) = 6.

In 1961 D. Rees [38] proved that in formally equidimensional local rings the

converse is also true.

THEOREM 1 (Rees, 1961). Let (A,m) be a formally equidimensional localring, let J ⊆ I be m-primary ideals such that e(J ) = e(I ). Then J is a reduction

of I .

For the geometric significance of this theorem we refer the reader to J. Lipman

[28].

If J ⊆ I are ideals in a local ring (A,m) such that J is a reduction of I , thene(J Ap) = e(I Ap) for each minimal prime ideal p of I . In general the converse is nottrue, but, using the characterization of ideals with height(I ) = s(I ) given by E. C. Dadein [12], E. Boger proved the following result.

THEOREM 2 (Boger [8], 1970). Let J ⊆ I be ideals in a formally equidimen-

sional local ring (A,m) such that√J =

√I , height(J ) = s(J ) and e(J Ap) = e(I Ap)

for each minimal prime ideal p of I . Then J is a reduction of I .

A further generalization of this result was given by B. Ulrich (see [16], (3.6.3)),

who proved:

PROPOSITION 1. Let (A,m) be a formally equidimensional local ring and J ⊆I be ideals of I . Then either height(J I n−1 : I n) < s(J ) for all n ≥ 1 or J is a

reduction of I .

COROLLARY 1. Let (A,m) and J ⊆ I as in the above proposition. Then J

is a reduction of I if and only if Jp is a reduction of Ip for all prime ideals p with

height(p) = s(Jp).

REMARK 1. If J ⊆ m is an ideal of a formally equidimensional local ring

(A,m), then {p ∈ Spec A | s(Jp) = height(p)} is a finite set, precisely, it coincideswith AssA(A/Jn) for sufficiently large n, where Jn denotes the integral closure of theideal Jn . In fact,

AssA(A/J ) ⊆ AssA(A/J 2) ⊆ AssA(A/J 3) ⊆ . . .


and this sequence eventually stabilizes to a limit set

∞⋃

k=1

AssA(A/J k) = AssA(A/Jn) (for some n > 0) ,

which is called the set of asymptotic primes of J (see [31], (4.1), p. 26) and which will

be denoted by Asymp(J ). One can prove that

Asymp(J ) = {p ∈ Spec A | ∃ P ∈ AssGJ (A)(GJ A) such that p = P ∩ A}= {p ∈ Spec A | s(Jp) = height(p)} .

3. Generalized Samuel multiplicities

This section is devoted to a generalization of Samuel’s multiplicity by a sequence of

numbers, the so-called generalized Samuel multiplicity, which we have introduced and

studied in several papers, partly in collaboration with H. Flenner. We also present the

main properties of this new multiplicity and its relation to other important invariants of

local rings.

3.1. Generalized Samuel multiplicities (see [4])

Let (A,m, k) be a d-dimensional local ring and let I ⊂ A be an arbitrary ideal (not

necessarily m-primary).

Let GI (A) :=⊕

j≥0 Ij/I j+1 be the associated graded ring of A with respect

to I and let us consider the bigraded ring

R =⊕

i, j≥0

Ri, j =⊕

i, j≥0

Gim(G

jI (A)) =

⊕

i, j≥0

(mi I j + I j+1)/(mi+1 I j + I j+1) ,

where R00 = A/m = k is a field.

Let H (0,0)(i, j) := dim Ri j the Hilbert function of the bigraded ring R and let

H (1,1)(i, j) :=j∑

q=0

i∑

p=0

H (0,0)(p, q)

its twofold sum transform. For both i, j >> 1 this function becomes a polynomial in

(i, j), which can be written in the form

∑

k+l≤da

(1,1)k,l

(i + k

k

)(j + l

l

).

Following [4] define the generalized Samuel multiplicity to be

(a(1,1)0,d , a

(1,1)1,d−1, . . . , a

(1,1)d,0 ) =: (c0(I ), c1(I ), . . . , cd(I )) =: c(I ) .


It will turn out that the first coefficient c0(I ) plays an important role as an intersectionnumber and permits generalizations of results about Samuel’s multiplicity. We will call

it the j -multiplicity j (I ) = j (I, A).

In a geometric way the j-multiplicity can be described as follows. Let X =Spec A, let d = dim X , let Y be the subscheme of X defined by I and let p : Z → X be

the blowing up of X along Y . Consider the union E of all the irreducible components of

the exceptional set p−1(Y ) contained in the special fiber p−1(m). This is a projectivescheme over A/mn for some n. The multiplicity j (I, A) is the (d − 1)-dimensionaldegree of E .

THEOREM 3 ([4], Prop. 2.3, 2.4, 2.5). With the above notations set q := dim

(A/I ), G := GI (A), s := s(I ) = dimG/mG. Then

1. ck = 0 for k < d − s and k > q, that is,

(c0, c1, . . . , cd) = (0, . . . , 0, cd−s, . . . , cq , 0, . . . , 0);

2. cd−s =∑

P e(mGP) · e(GP),

where P runs over all highest dimensional associated prime ideals of G/mGsuch that dimG/P + dimGP = dimG;

3. cq =∑

p e(I Ap) · e(A/p),

where p runs over all highest dimensional associated prime ideals of A/I suchthat dim A/p + dim Ap = dim A;

4. e(GI (A)) =∑d

k=0 ck(I, A);

5. if height(I ) = s(I ) then

(c0, . . . , cq−1, cq , cq+1, . . . , cd) = (0, . . . , 0, cq , 0, . . . , 0).

In particular, if I is m-primary, then height(I ) = s(I ) = d, q = 0 and

(c0, . . . , cd) = (e(I ), 0, . . . , 0) ,

that is, the sequence (c0, . . . , cd) generalizes the Samuel multiplicity to arbitrary ide-als.

REMARK 2. By Theorem 3, (1) and (2), if G = GI A is formally equidimen-

sional, then

j (I ) = c0(I ) /= 0 if and only if s(I ) = dim A .

REMARK 3. With the notation of (3.1) N.V. Trung (see [46], Cor. 2.8) provedthat if (A,m) and I are such that the ring R = Gm(GI A) is a domain or a Cohen–Macaulay ring, then ci (I ) > 0 for all d − s ≤ i ≤ q where d = dim A, s = s(I ) andq = dim(A/I ).


REMARK 4. If J ⊂ I ⊂ A are two ideals of A with√I =

√J and c(I ) =

(c0(I ), . . . , cd(I )), c(J ) = (c0(J ), . . . , cd(J )) are their generalized Samuel multiplic-ities, then from Theorem 3 (3) we have cq(I ) ≤ cq(J ), but we can say nothing aboutcq−1(I ) and cq−1(J ), as one can see from the two following examples.

EXAMPLE 2. (Erika Giorgi) Let A = k[x, y, z](x,y,z) where K is a field and

consider the ideals J = (x, y)3 ∩ (x, z)3 ∩ (y, z) and I = (x2, y) ∩ (x, z)2 ∩ (y, z).Obviously J ⊂ I and

√J =

√I . By using [1] we get c(J ) = (4, 19, 0, 0) and

c(I ) = (6, 7, 0, 0).

Let L = (x2, y2)∩(x3, z3)∩(y, z) and M = (x, y)2∩(x, z)3∩(y, z). ObviouslyL ⊂ M and

√L =

√M . In this case we get c(L) = (29, 14, 0, 0) and c(M) =

(0, 14, 0, 0).

The last example shows also the importance of the condition height(L) = s(L)in Boger’s Theorem 2. Here the condition e(Lp, Ap) = e(Mp, Ap) for all minimalprimes p of M is satisfied, but L is not a reduction of M . In fact, if L was a reduction

of M then L and M would share a minimal reduction and then they would have the

same analytic spread, but s(L) = dim A = 3, while s(M) = 2.

3.2. Generalized Samuel multiplicities as intersection numbers

Let X , Y be equidimensional closed subschemes of PnK = Proj(K [X0, . . . , Xn]),where K is an arbitrary field. For indeterminates Ui j (0 ≤ i, j ≤ n) let L be the

pure transcendental field extension K (Ui j )0≤i, j≤n and XL := X ⊗K L , etc. Proving

a Bezout theorem for improper intersections, Stuckrad and Vogel (see [16]) introduced

a cycle v(X,Y ) = v0 + · · · + vn on XL ∩ YL , which is obtained by an intersection

algorithm on the ruled join variety

J := J (XL ,YL) ⊂ P2n+1L = Proj(L[X0, . . . , Xn,Y0, . . . ,Yn])

as follows.

Let ! be the “diagonal” subspace of P2n+1L given by the equations

X0 − Y0 = · · · = Xn − Yn = 0 ,

let Hi ⊆ J be the divisor given by the equation

"i :=n∑

j=0

Ui j (X j − Y j ) = 0,

and set " := ("0, . . . , "n). Then one defines inductively cycles βk and vk by settingβ0 := [J ]. If βk is already defined, decompose the intersection

βk ∩ Hk = vk+1 + βk+1 (0 ≤ k ≤ dim J ) ,

where the support of vk+1 lies in ! and where no component of βk+1 is containedin ! (if k = dim J , then the support of vk+1 is the empty set). It follows that vk is


a (dim J − k)-cycle on XL ∩ YL ∼= J ∩ !. The part of dimension k of the cyclev(X,Y ) := v(", J ) :=

∑vk will be denoted by vk , so that the upper index denotes the

codimension in the ruled join and the lower one the dimension of the cycle. In general,

the cycle v(X,Y ) is defined over L .

DEFINITION 1. The cycle v(X,Y ) is called the v-cycle of the intersection of Xand Y . An irreducible subvariety C of XL ∩ YL is said to be a characteristic subvarietyif C occurs in v(X,Y ). The coefficient of C in v(X,Y ) is denoted by j (X,Y ;C). Thus

v(X,Y ) =∑

C

j (X,Y ;C) [C] ,

where C runs through the characteristic subvarieties. The set of all these subvarieties is

denoted by C = C(X,Y ). Moreover, the set of all elements of C which are defined overK is denoted by Crat = Crat (X,Y ), that is, Crat is the set of K -rational or distinguishedor fixed subvarieties and C \ Crat is the set of the so-called non K -rational or movable

subvarieties of the intersection of X and Y .

By a result of van Gastel ([21], Prop. 3.9), a K -rational irreducible subvariety

C of XL ∩ YL occurs in v(X,Y ) if and only if C is a distinguished variety of the

intersection of X and Y in the sense of Fulton ([18], p. 95), and this is equivalent to the

maximality of the analytic spread (see [2]) or the maximality of the dimension of the

so-called limit of join variety (see [17]).

For an arbitrary irreducible subvariety Z ⊆ XL ∩ YL ⊂ PnL we set Z! :=J (Z , Z) ∩ !. By J and Z! we denote the affine cones of the ruled join J :=J (XL ,YL) ⊂ P2n+1L and Z! in the affine space A2n+2L . Let (A,m) be the local ringOJ ,Z!

and I ⊂ A be the ideal of the diagonal subspace ! and let G(X,Y ; Z) denote

the associated graded ring GI (A) = ⊕∞j=0 I

j/I j+1. If Z is the empty subvariety of

Pn , then A becomes the homogeneous ring of coordinates of the ruled join J ⊂ P2n+1L

localized at the irrelevant maximal ideal; that is, we obtain a global picture of the in-

tersection algorithm.

PROPOSITION 2 ([4], Section 4). With the preceding notation,

e(G(X,Y ; Z)) = e(GI (A)) =d∑

k=0

ck(I, A) =∑

C

j (X,Y ;C) · e(OC,Z ) ,

where C runs through the characteristic subvarieties of X and Y with C ⊇ Z. In

particular, if Z ∈ Crat (X,Y ), then j (X,Y ; Z) = j (I, A).

If Z = ∅, then d = dim A = dim J + 1 and

c0 = j (X,Y ;∅), c1 = deg v0, c2 = deg v1, . . . , cd = deg vd−1 ;

moreover, if k > dim(X ∩ Y ) + 1, then ck = 0.

If Z /= ∅ is K -rational, then d = dim A = dim J − dim Z and

ck =∑

C

j (X,Y ;C) · e(OC,Z ) (0 ≤ k ≤ d) ,


where C runs through all varieties of C(X,Y ) with C ⊇ Z and codimC Z = k. If

k > dim(X ∩ Y ) − dim Z, then ck = 0.

We will illustrate the above proposition by recomputing the self-intersection of

a monomial curve in P3 obtained by hand in [44], Example 2, p. 269, as a result of aheavy calculation.

EXAMPLE 3. Consider the curve X in P3K (char(K ) /= 2, 3) given parametri-

cally by (s6, s4t2, s3t3, t6) and with defining ideal

I (X) = (x0x3 − x22 , x20 x3 − x31) ⊂ K [x0, x1, x2, x3] .

Using [1] (see the sample file Segre4.txt) and the above proposition in the case Z =∅, we want to calculate its self-intersection cycle v(X, X). Running the computerprogram, with the notation of the proposition, we get c(I, A) = (c0, c1, c2, 0, 0), where

c0(I, A) = j (X, X;∅) = 12,

c1(I, A) = deg(v0(X, X)) = 18,

c2(I, A) = deg(v1(X, X)) = 6.

Since X is a complete intersection of degree 6, it follows that j (X, X; X) = 1. In

order to understand v1(X, X) we recall that by [2], Corollary 2.5, a point of X is a

K -rational component of v1(X, X) if and only if it is a singular point of X . One checksthat X has the two singular points P = (0 : 0 : 0 : 1) and Q = (1 : 0 : 0 : 0).One applies now Proposition 2 in the cases Z = P and Z = Q, more precisely, in the

previous calculation one substitutes x3 = 1 for P obtaining c = (8, 3, 0) and x0 = 1

for Q obtaining c = (3, 2, 0). This means that P is a point of multiplicity 3 in X andj (X, X; P) = 8 and Q is a double point of X and j (X, X, Q) = 3. The contribution

of non K -rational points is therefore 7.

REMARK 5 (Analytic case). In the paper [47], Tworzewski has constructed an

intersection cycle for complex analytic subsets X and Y of a manifold M which do

not intersect necessarily properly. His construction is based on a pointwise defined

intersection multiplicity g(x) = g(X × Y,!M , x) for a point x ∈ !M , where !M is

the diagonal of M×M and g(x) is the sum of the coordinates of the so-called extendedindex of intersection g(x) (see [47], Definition (4.2), p. 185).

Let A = OX×Y,x and I = I!M·OX×Y,x . K. Nowak [35], [36] (see also [6]) has

proved that g(x) = e(GI (A)) and that g(X) is composed of the generalized Samuelmultiplicities c0(I, A), . . . , cdim(X∩Y )(I, A) and of zeros.

REMARK 6. Recall that a d-dimensional projective variety X is said to be con-

nected in dimension d − 1 if for every closed subvariety Z of X of dimension < d − 1the set X \ Z is connected.

Flenner, van Gastel and Vogel (see [13], Theorem 3.4) proved that if X and

Y are pure dimensional projective varieties connected in dim X − 1 and dim Y − 1

respectively, A is the ring of coordinates of the ruled join of X and Y localized at the


irrelevant maximal ideal and I is the ideal of “the diagonal” in the ring A, then we have

ci (I ) > 0 for all d − s ≤ i ≤ q, where d = dim A, s = s(I ) and q = dim(A/I ).

The assumption of the theorem by Flenner–van Gastel–Vogel does not imply

the assumption of Trung’s Corollary 2.8 as one can see by the following example.

EXAMPLE 4. ([42]) Let X ⊂ P3K be the non-singular curve of F. S. Macau-

lay ([29], page 98) given parametrically by {(s4, s3t, st3, t4)} and let Y ⊂ P3 be theline x0 = x1 = 0. Then X and Y are connected in dimension 0 and the theorem of

Flenner–van Gastel–Vogel can be applied.

However, with the notation of Remark 6, the ring R = Gm(GI (A)) is nei-ther Cohen–Macaulay (since the coordinate ring of X is not Cohen–Macaulay) nor a

domain (since the intersection cycle v(X,Y ) has two K -rational components), henceboth the conditions of Trung [46], Corollary 2.8 are not fulfilled.

3.3. The j-multiplicity as a length

We have seen that if (A,m) is a local ring of dimension d and I ⊂ A is an arbitrary

ideal, the j-multiplicity j (I ) := c0(I ) is an important generalization of the classicalSamuel multiplicity of an m-primary ideal since it measures the contribution of distin-guished components of the intersection (see Proposition 2).

If the ring A is Cohen–Macaulay and the ideal I is m-primary, then e(I, A) isgiven by

length(A/( f1, . . . , fd))

where f1, . . . , fd ∈ I are sufficiently generic elements, hence a minimal reduction of

I . Using the theory of residual intersections due to Huneke (see [25], [26]) and others,

under certain hypothesis on the pair (A, I ) a similar formula can be proved for thej-multiplicity, (see [15], Theorem 3.4).

We recall the following definitions. Let A be a local ring, I ⊆ A an ideal.

As usual H∗(I, A) will denote the Koszul cohomology of (I, A), i.e. H∗(I, A) is thecohomology of the Koszul complex K •(x1, . . . , xk; A), where x1, . . . , xk ∈ I is a

minimal set of generators for I . Following [16], the pair (A, I ) is called stronglyCohen–Macaulay (SCM) if H p(I, A) is either zero or a Cohen–Macaulay module forall p ≥ 0; note that this differs from the notion originally given in [25] as we also

require the Cohen–Macaulayness of A. For basic properties of this concept we refer

the reader to [25] and [16] (7.2). In particular we will need the following two facts.

If (A, I ) is SCM and H p(I, A) is nonzero then it is automatically a Cohen–Macaulay module of dimension dim A/I over A/I , see [25] or [16], (7.2.7). Moreover,the pair (A, I ) is SCM if and only if the Koszul cohomology H∗(y1, . . . , ye; A) is aCohen–Macaulay module for an arbitrary generating set y1, . . . , ye of I .

Another important notion in the theory of residual intersections is the Artin–

Nagata condition. An ideal I of a local ring A is said to satisfy the Artin–Nagata

condition Gs if

(Gs) µ(Ip) ≤ height p for all primes p ∈ V (I ) with height p < s.


Here µ(Ip) denotes the minimal number of generators of the ideal Ip ⊆ Ap. The ideal

I is said to satisfy G∞ if Gs holds for all s ≥ 1.

THEOREM 4 ([15], Theorem 3.4). Let A be a d-dimensional local Cohen–

Macaulay ring, I ⊆ A an ideal. Assume that (A, I ) is SCM and that I satisfies Gd.

Then j (I, A) is given by the length of

I/( f1 + . . . + fd−1) + fd I,

where f1, . . . , fd are sufficiently generic elements of I .

REMARK 7. For the precise meaning of “sufficiently generic elements of I”

see [15], (3.3). Let I = (x1, . . . , xn). Extending the ring A with new indeterminatesu11, . . . , udn , that is passing from A to A′ = A[u11, . . . , udn]mA[u11,...,udn ], the ele-ments

fi :=n∑

h=1

uihxh for i = 1, . . . , d

are sufficiently generic.

REMARK 8. If I is an m-primary ideal of a d-dimensional Cohen–Macaulayring A, then (A, I ) is automatically strongly Cohen–Macaulay and satisfies the Artin–Nagata condition Gd , so Theorem 4 generalizes the classical length formula.

REMARK 9. If one admits the assumption on the local ring A and I only sat-

isfies the Gd -condition, then by the proof of [15], Theorem 3.4 one can see that the

following inequality holds:

j (I, A) ≤ length(I/( f1, . . . , fd−1) + fd I ) ,

which generalizes the classical inequality for the Samuel multiplicity of a system of

parameters given in (2.3).

The result of Theorem 4 can be applied to give explicitly expressions for the

j-multiplicity in many examples where the SCM condition is satisfied (see [15], Sec-

tion 4). In particular one obtains a positive answer to the following problem posed by

Ein, Lazarsfeld and Nakamaye in some special cases.

PROBLEM 1. Let H ⊆ An be a hypersurface and C ⊆ H an irreducible subset

of codimension c such that C is an irreducible component of the two equimultiplicity

strata $l and $l+m , where $i := {x ∈ H | e(OH,x ) > i}.Is then C a distinguished component of the (c + 1)-fold intersection Hc+1?

Moreover, does C appear with a coefficient ≥ mc+1 in the intersection cycle?

The general problem is even open in the special case when C is a point where

the multiplicity jumps (see [15], (4.4)). In this case the question becomes: assume that

H ⊆ An is a hypersurface with a jump of multiplicity at 0, i.e. e(OH,0) > e(OH,x )


for all x /= 0 near 0. Is the point 0 then a distinguished component of the n-fold self-

intersection of H? By [15], Prop. 4.1 in this situation we only know that C = 0 is

a distinguished component of Hn when it is an irreducible component of the singular

locus Sing H .

3.4. Intersection algorithms for filter-regular sequences and computation of the

j-multiplicity

In [3] and [4] we introduced intersection algorithms in a local ring, which are counter-

parts of the construction of the Stuckrad–Vogel cycle (see Definition 1), and compared

them with analogous algorithms in the associated graded ring. These algorithms can

be used to express multiplicities as lengthes and to generalize the last result of 2.3.

Again let (A,m) be a d-dimensional local ring, let I ⊆ m be an ideal of A and

letG := GI (A) be the associated graded ring. Consider a sequence a = (a1, . . . , at ) ofelements of I such that

√aA =

√I and the sequence a∗ = (a∗1 , . . . , a

∗t ) of the initial

forms of a1, . . . , at in G is contained in G1 = A/I and is a filter-regular sequencewith respect to the ideal G+ = ⊕n≥1 I

n/I n+1, that is

(a∗1 , . . . , a∗k−1)G :G a∗k ⊆

⋃

n>0

((a∗1 , . . . , a∗k−1)G :G (G+)n)

for k = 1, . . . , t , or equivalently, a∗k /∈ P for all relevant associated prime ideals

P ∈ AssG(G/(a∗1 , . . . , a∗k−1)G) for k = 1, . . . , t (see, for example, [43], Def. 1,

p. 252). In particular this implies that a = (a1, . . . , at ) is a filter-regular sequence in Awith respect to I , see, for example, [3], (2.2).

We define a cycle v(a, A) of A supported on V (I ) = V (aA) by the followingintersection algorithm in A. Set a−1 := (0), a0 := 0, J := aA and inductively

ak :=⋃

n≥0

((ak−1 + ak A) :A Jn) (0 ≤ k ≤ t) .

Observe that at = A. Then

vk(a, A) :=∑

p

length(A/(ak−1 + ak A))p [p] ,

where the sum is taken over all (d − k)-dimensional associated prime ideals p ofA/(ak−1 + ak A) that contain J and [p] denotes the cycle associated with p. We definev(a, A) :=

∑tk=0 vk(a, A), and the degree of vk(a, A) by

deg vk(a, A) :=∑

p

length(A/(ak−1 + ak A))p · e(A/p) .

The cycle v(a, A) can also be constructed by the following unmixed intersectionalgorithm in A, which is more closely related to the approach of Stuckrad and Vogel in


[42]. Recall that, given an ideal L , we denote by U (L) the intersection of all highestdimensional primary ideals of L . Set a′−1 := (0) and inductively

a′k :=⋃

n≥0

(U (a′k−1 + ak A)) :A Jn) (0 ≤ k ≤ t) .

Then, if a′k /= A, it holds a′k = U (ak) (see [3], proof of Proposition 3.2), hence

vk(a, A) =∑

p

length(A/(a′k−1 + ak A))p [p] ,

where the sum is taken over all (d − k)-dimensional associated prime ideals p ofA/(a′k−1 + ak A) that contain J .

In the same way, replacing a by a∗ and J by G+, we define a cycle v(a∗,G) byan intersection algorithm in G = GI (A) with a−1 := 0 · G, a∗0 := 0, and

ak := (ak−1 + a∗k G) :G 〈G+〉 (0 ≤ k ≤ t) .

We put

vk(a∗,G) :=

∑

P

length(G/(ak−1 + a∗k G))P [P] ,

where the sum is over all (d− k)-dimensional associated prime idealsP of G/(ak−1+a∗k G) that contain G+. Observe that the prime ideals of v(a; A) contain I and hencecorrespond to prime ideals in the ring A/I . On the other hand, the prime ideals ofv(a∗,G) contain G+ and correspond to their contraction ideals in G0I (A) = A/I . Soboth cycles v(a; A) and v(a∗,G) can be considered as cycles of A/I and we have thefollowing theorem ([4], 3.3):

THEOREM 5 (Deformation to the normal cone).

v(a, A) = v(a∗,G) as cycles of A/I .

A natural deformation space for the deformation to the normal cone is given by

the extended Rees ring of A with respect to I . L. O’Carroll and T. Pruschke used this

to introduce an analogue of the previous algorithms in Rees rings, see [37].

In order to generalize the last result of 2.3, that is, to compute the j-multiplicity

as a length, we must use in the above algorithms “generic elements” a1, . . . , ad of I .The precise meaning of “generic” is that the elements must be a “super-reduction” in

the sense of [3], (2.7):

DEFINITION 2. Let (A,m) be a local ring, let I be an ideal of A such that

s(I ) = dim A = d. A sequence of elements a1, . . . , ad in I is called a super-reductionfor I if:

1. their initial forms a∗1 , . . . , a∗d in G = GI (A) are of degree one and form a

filter-regular sequence for G with respect to G+ :=⊕

i>0 Ii/I i+1, that is,

(a∗1 , . . . , a∗i−1)G : a∗i ⊆

⋃n≥0((a

∗1 , . . . , a

∗i−1)G : (G+)n);


2. for every relevant highest dimensional prime ideal p of G = GI A and d(p) :=dimG/(mG + p) the initial forms a∗1 , . . . , a

∗d(p) are a system of parameters for

G/(mG + p).

REMARK 10. If A/m is infinite, then every ideal of maximal analytic spread

has a super-reduction ([3], (2.9)).

If (a1, . . . , ad) is a super-reduction of I , then a1, . . . , ad form a minimal basisof a minimal reduction of I (see [3], (2.8)).

If I is m-primary, then the notion of super-reduction coincides with that of asuperficial system of parameters in the sense of [39], p. 185.

Now let (A,m) be a d-dimensional local ring, let I be an ideal of A of maximalanalytic spread s(I ) = dim A = d > 0 and let a = (a1, . . . , ad) be a super-reductionof I . We set

Int(a, A) := ad−1 + ad A and U-Int(a, A) := a′d−1 + ad A,

where ad−1 and a′d−1 are the ideals produced by the intersection algorithm and the un-mixed intersection algorithm, respectively. Then the ideals Int(a, A), U-Int(a, A), areequal andm-primary (see [3], 3.2). Analogously, the ideals Int(a∗,G) and U-Int(a∗,G)are equal and primary with respect to the homogeneous maximal ideal of G (see [3],

3.3) and we have the following theorem ([3], Theorem 3.8):

THEOREM 6 (Computation of the j-multiplicity by super-reductions). Let

(A,m) be a d-dimensional local ring, let I be an ideal of A of maximal analytic spreads(I ) = dim A = d > 0 and let a = (a1, . . . , ad) be a super-reduction of I . With thenotation introduced before, one has

j (I, A) = lengthA(A/ Int(a, A)) = lengthA(A/U-Int(a, A))

= lengthG(G/ Int(a∗,G)) = lengthG(G/U-Int(a∗,G)) = j (G+,G).

3.5. Reduction ideals and j-multiplicity

Using the j-multiplicity it is possible to give a numerical characterization of reduction

ideals, which generalizes Boger’s theorem, see Theorem 2. The easy direction is given

by the following proposition.

PROPOSITION 3 ([14], Proposition 2.10). Let (A,m) be a local ring, let J ⊆I ⊆ m be ideals of A. If J is a reduction of I , then j (J, A) = j (I, A).

For the other direction we need to consider formally equidimensional local

rings. Precisely we have the following result.

THEOREM 7 ([14], Theorem 3.3). Let J ⊆ I ⊆ m be ideals of an equidi-

mensional local ring (A,m). Then J is a reduction of I if and only if j (Jp, Ap) =j (Ip, Ap) for all prime ideals p ∈ Spec A.


REMARK 11. Theorem 7 generalizes Boger’s theorem (see Theorem 2), since if√I =

√J then the ideals I and J have the same minimal primes, that is Min(A/I ) =

Min(A/J ). Moreover s(J ) = height(J ) implies that

{p ∈ Spec A | height(p) = s(Jp)} = Asymp(J ) = Min(A/I )

by Lipman’s Theorem 3, p. 116 and remark p. 117.

For each p ∈ Min(A/I ) we have

j (Jp, Ap) = e(Jp, Ap)

j (Ip, Ap) = e(Ip, Ap)

hence by Corollary 1 of (2.4) and Rees’ theorem (see Theorem 1) we can conclude that

J is a reduction of I .

REMARK 12. The numerical condition of Theorem 7 must be checked only

for a finite number of prime ideals, p ∈ Spec A. In fact, by Remark 2 in (3.1) and

Remark 1 in (2.4) the multiplicity j (Jp, Ap) /= 0 if and only if p ∈ Asymp(J ), henceit is sufficient to compare j (Jp, Ap) and j (Ip, Ap) for all p ∈ Asymp(J ) ∪ Asymp(I )and this is a finite set.

REMARK 13. Theorem 7 says that I is the largest ideal N containing I such

that j (Ip, Ap) = j (Np, Ap) for all p ∈ Spec A, see (2.4).

The proof of Theorem 7 uses the generalized multiplicity j (I,M) for a finiteA-module M , which was introduced in [16], Section 6.1. This multiplicity coincides

with j (I, A) when M = A, has nice properties like additivity for exact sequences of

A-modules (see [15], Lemma 3.1) and it is preserved under generic hyperplane sections

(see [15], Proposition 3.2). These properties are used to prove Theorem 7.

In [10] C. Ciuperca generalizes Proposition 3 in the following way.

PROPOSITION 4 ([10], Proposition 2.7). Let (A,m) be a d-dimensional localring and let J ⊆ I ⊆ m be ideals of A. If J is a reduction of I , then J and I have the

same generalized Samuel multiplicities, that is c(J ) = c(I ).

It would be interesting to have a converse of this proposition, which is known

in the analytic case, see [19], Corollary 4.9 and our Thorem 11. This would avoid

localization and would give a more useful numerical condition to test if an ideal J is a

reduction of I by using computer algebra systems.

3.6. Generalized Hilbert coefficients and Serre’s property (S2) for the Rees alge-bra

Let (A,m) be a formally equidimensional local ring of dimension d, let I be an idealin A, let R = A[I t, t−1] be the extended Rees algebra of A with respect to I and


let R = ⊕n∈Z I ntn be the integral closure of R. Then Flenner-Manaresi’s numerical

characterization of reduction ideals of Theorem 7 can be interpreted as follows:

I n is equal to the largest ideal, say K , containing I n such that j (I n Ap, Ap)= j (Kp, Ap) for all p ∈ Spec A.

Let R be a noetherian integral domain. Then the well-known normality criterion

of Krull-Serre states that R is integrally closed if and only if it satisfies the following

two properties:

(R1) for each prime P of codimension ≤ 1, RP is regular;

(S2) for each prime P of codimension ≥ 2, depth RP ≥ 2.

If R is lacking the S2-property of Serre, one can try to construct the minimal extension

R of R which satisfies S2. The ring R is called the S2-closure or the S2-ification of

R and it exists when R has a canonical module or when R is a universally catenary,

analytically unramified domain (see [22], (5.11.2) and [24], (2.7)). It is a step in the

construction of the integral closure of R.

If I ⊂ A is anm-primary ideal, then K. Shah [41] proved the existence of uniquelargest ideals (the so-called k-th coefficient ideals) I{k} (1 ≤ k ≤ d) lying between Iand I such that the first k + 1 Hilbert coefficients e0, . . . , ek (see Section 2.1) of Iand I{k} coincide. If A is a formally equidimensional, analytically unramified localdomain with infinite residue field and if A has positive dimension and is (S2), thenC. Ciuperca [9] showed that the n-th graded piece of the S2-closure of R = A[I t, t−1]is precisely the first coefficient ideal (I n){1}, that is, the largest ideal K ⊇ I n such that

e0(K ) = e0(In) and e1(K ) = e1(I

n).

Using the generalized Hilbert coefficients a(1,1)k,l (see Section 3.1), Ciuperca [10]

has generalized this to not necessarily m-primary ideals I . In order to describe Ciu-perca’s result, we assume that d = dim A > 0 and introduce the following notation:

j0(I ) := j (I ) = c0(I ) = a(1,1)0,d (I ) ,

j1(I ) := (c1(I ), a(1,1)0,d−1(I )) = (a

(1,1)1,d−1(I ), a

(1,1)0,d−1(I )) .

Note that in the case of an m-primary ideal I one gets the first two classical Hilbertcoefficients: j0(I ) = e0(I ) = e(I ) and j1(I ) = (0,−e1(I )). Ciuperca ([10], Def. 3.1)extended the definition of the first coefficient ideal I{1} to not necessarily m-primaryideals I as follows: if dim A/I < dim A, she defined

I{1} :=⋃

(I n+1 :A a) ,

where the union ranges over all n ≥ 1 and all a ∈ I n \ I n+1 such that the initial forma∗ of a in GI (A) is a part of a system of parameters of GI (A). If I is m-primary, thisdefinition coincides with the one given by Shah. Indeed, by the structure theorem for

the coefficient ideals proved by Shah ([41], Theorem 2), we have I{1} =⋃

(I n+1 :A a),where the union ranges over all n ≥ 1 and all a ∈ I n extendable to some minimal

reduction of I n . Note that a is extendable to some minimal reduction of I n if and only


if the image of a∗ in GI (A)/mGI (A) is part of a system of parameters. But if the idealI is m-primary this is equivalent to the fact that a∗ is part of a system of parameters ofGI (A), since the ideal mGI (A) is nilpotent.

With this new definition of the first coefficient ideal I{1}, the above descriptionof the S2-closure of the extended Rees algebra can be generalized to not necessarily

m-primary ideals I .

THEOREM 8 ([10], Theorem 3.4). Let (A,m) be a formally equidimensional,analytically unramified local domain with infinite residue field and positive dimension,

and let I be an arbitrary ideal of A. If R = ⊕n∈Z Intn is the S2-ification of R =

A[I t, t−1], thenIn ∩ A = (I n){1} for all n ≥ 1 .

In particular, if A is (S2), then In = (I n){1} for all n ≥ 1.

Now the announced numerical characterization of the S2-ification of the ex-

tended Rees algebra reduces to the problem of finding a numerical characterization of

the generalized first coefficient ideals. This is the contents of the following theorem.

THEOREM 9 ([10], Theorem 4.5). Let (A,m) be a formally equidimensionallocal ring and let J ⊆ I ⊆ m be ideals of positive height. Then the following condi-

tions are equivalent:

1. I ⊆ J{1};

2. j0(J Ap) = j0(I Ap) and j1(J Ap) = j1(I Ap) for all p ∈ Spec A;

3. j0(J Ap) = j0(I Ap) and a(1,1)0,d−1(I Ap) = a

(1,1)0,d−1(J Ap) for all p ∈ Spec A.

REMARK 14. Condition 3 of the previous theorem is not contained in [10],

Theorem 4.5. Obviously 2 implies 3. To see the converse, one can observe that by

Theorem 7 from j0(J Ap) = j0(I Ap) for all p ∈ Spec A it follows that J is a reductionof I , hence Jp is a reduction of Ip for all p ∈ Spec A, therefore by Proposition 4 onehas c1(J Ap) = c1(I Ap).

In view of the above consideration it seems to be better to define j1(I ) :=a

(1,1)0,d−1(I ) instead of j1(I ) := (c1(I ), a

(1,1)0,d−1(I )).

The previous theorem can be considered as a generalization of Flenner-Manaresi’s nu-

merical characterization of reduction ideals (Theorem 7), which can be reformulated

as follows.

THEOREM 10. Let (A,m) be a formally equidimensional local ring and letJ ⊆ I ⊆ m be ideals of A. Then the following conditions are equivalent:

1. I ⊆ J{0} := J , that is, J is a reduction of I ;

2. j0(I Ap) = j0(J Ap) for all p ∈ Spec A;


3. j0(I Ap) = j0(J Ap) for all ideals p ∈ Asymp(I ) ∪ Asymp(J ).

PROBLEM 2. In the previous theorem only a finite number of localizations has

to be considered, see Remark 12 in (3.5). It is not clear if it is enough to test also

condition 2 of Theorem 9 only for a finite number of prime ideals p ∈ Spec A.

3.7. Generalized Samuel multiplicities and Segre numbers

T. Gaffney and R. Gassler [19] introduced and studied the so-called Segre numbers of

an ideal in the following set-up.

Let (X, 0) ⊆ (Cn, 0) denote a germ of an analytic subset of pure dimension

d, and let I be an ideal in OX,0 which defines a nowhere dense subspace of (X, 0).Choose a minimal set h1, . . . , hr of generators of I . The polar varieties Pk(I, X) andSegre cycles %k+1(I, X) are defined inductively for k = 0, . . . , d − 1 as follows (see[19] and [20], Section 2.1): P0(I, X) := X , and for k ≥ 1 the polar variety Pk(I, X)is defined to be the closure of V (hk |Pk−1(I,X)) \ V (I ), where hk is a generic linearcombination of h1, . . . , hr . The word “generic” means in particular that the subspaceY of Pk−1(I, X) defined by the sheaf of ideals (hk)OPk−1(I,X) has to be reduced outside

V (I ) in a sufficiently small neighbourhood of the point 0. The k-th Segre cycle isdefined as the difference of cycles

%k(I, X) := [V (hk |Pk−1(I,X))] − [Pk(I, X)] .

We recall that the cycle [V (hk |Pk−1(I,X)] is defined as∑mW [W ], where the W ’s run

over all irreducible components of the set V (hk |Pk−1(I,X)), and the integer mW equals

by definition the length of the local ring (OY,y)(Wy)i, where (Wy)i is the prime ideal of

a component of the germ of the set W at a point y ∈ W (see [30], p. 9).

The k-th Segre cycle %k(I, X) can also be described by using the blowup of Xalong V (I ). Let

X × Pr−1 ⊃ BlI (X)b→ X ,

E the exceptional divisor and H1, . . . , Hk−1 generic hyperplanes on BlI (X) inducedby generic hyperplanes of Pr−1. Then

%k(I, X) = b∗(H1 · · · Hk−1 · E · BlI X) .

The k-th Segre number is defined as

ek(I, X) := mult0(%k(I, X)) := e(O%k (I,X),0), k = 1, . . . , d.

If s denotes the analytic spread of I , then one can easily see that the sequence

h := (h1, . . . , hs) is filter-regular with respect to I , hence one can perform the in-

tersection algorithm for the sequence h as in Section 3.6. By the definition of Segre

cycles, one has

mult0(%k(I, X)) = deg vk(h,OX,0) for k = 1, . . . , s, and

mult0(%k(I, X)) = 0 for k = s + 1, . . . , d


(see [6], proof of Theorem 1).

THEOREM 11 ([6], Theorem 2). With the previous notation, the following equal-

ities hold:

ek(I, X) = cd−k(I,OX,0) for k = 1, . . . , d

and cd(I,OX,0) = 0.

The generalized Samuel multiplicities are also related to the degrees of Segre

classes of cones and subvarieties Y ⊂ X . For the theory of Segre classes we refer the

reader to [18], Chapter 4.

The total Segre class s(Y, X) ∈ A∗Y is defined as follows: if Y = X then

s(Y, X) = [X ], otherwise let X = BlY X , C := CY X the normal cone of X along

Y , E = P(C) the exceptional divisor, η : E → Y the projection, and d := dim X =dim X . The i-fold self intersections Ei = E ∗ · · · ∗ E are well defined classes in

Ad−i (E) and one defines

s(Y, X) :=∑

i≥1

(−1)i−1η∗(Ei ) .

This means that the total Segre class is constructed by blowing up X along Y , and

pushing down various self-intersections of the exceptional divisor. It depends only on

the normal cone to Y in X . One writes

s(Y, X) = s(CY X) :=∑

i≥0

si (Y, X) =∑

i≥0

si (Y, X) ,

where by si one denotes the part of s of dimension i , and by si the part of codimension

i in X . Thus, if X is equidimensional (as we always assume), then si = sdim X−i .

If X and Y are nonsingular, then the normal cone is a bundle, the normal bundle

NY X of X along Y , with Chern classes ci = ci (NY X), and the Segre classes si (NY X)can be regarded as their formal inverse:

1+ c1 + c2 · · · = (1+ s1 + s2 + · · · )−1 ,

i. e., s0 = 1, s1 = −c1, s2 = c21 − c2 , . . .

Turning back to the general case, that is, X and Y not necessarily nonsingular,

for E = P(C) one has that NE X = OX(E)|E = OC (−1) is the dual of the canonical

line bundle OC on P(C). It follows that

Ei = (−1)i−1c1((OC (1))i−1) ∩ [P(C)] ,

hence

s = s(C) = s(Y, X) =∑

i≥1

η∗(c1(OC (1))i−1 ∩ [P(C)]) .

If Y ⊂ X ⊆ Pn is an irreducible and reduced subscheme of X and r :=codimXY > 0, q := dim(X) + 1 − r , then the degree of the Segre class sr (Y, X) =


sr (CY X) is related to the Samuel multiplicity e(OX,Y ) = eY X of X along Y as follows

(see [18], 4.3):

e(OX,Y )[Y ] = eY X [Y ] = sr (Y, X) = η∗(c1(OC (1))r−1 ∩ [P(C)])= (−1)r−1η∗(Er−1),

that is,

deg sr (CY X) = deg sq−1(CY X) = e(OX,Y ) · deg Y = cq(I, A) ,

where A is the homogeneous ring of coordinates of X localized at the irrelevant maxi-

mal ideal and I is the ideal of Y in A.

In general, with the convention that(m−1)

:= 0 for m ≥ 0 and(−1−1)

:= 1, one

has the following proposition, which gives the relation between generalized Samuel

multiplicities and degrees of Segre classes of cones.

PROPOSITION 5 ([4], Corollary 4.3). Under the hypothesis of Proposition 2, if

Z = ∅ then d = dim A = dim J +1, q = dim(J ∩!)+1 and, for k = −1, . . . , d−1,

ck+1(I, A) =q−1∑

i=k

(d − k − 2d − i − 2

)deg si (CJ∩! J )

and

deg sk(CJ∩! J ) = deg sd−k−1(CJ∩! J ) =k∑

i=0

(k − 1i − 1

)(−1)k−i cd−i (I, A) .

REMARK 15. More general, if X is an equidimensional algebraic scheme over

the base field K , L a line bundle of degree δ on X , σ1, . . . , σt ∈ H0(X,L) and Y :=V (σ1) ∩ · · · ∩ V (σt ), then

(3) ck+1 =q−1∑

i=k

(d − k − 2d − i − 2

)δi−k deg si (Y, X)

and

(4) deg sk(Y, X) = deg sd−k−1(Y, X) =k∑

i=0

(k − 1i − 1

)(−δ)k−i cd−i ,

k = 0, . . . , d − 1.

We want to illustrate the usefulness of the generalized Samuel multiplicities

for the calculation of the degrees of Segre classes using computer algebra systems,

discussing an example which can be easily checked by hand. The same method can be

applied to much more complicated examples which cannot be calculated by hand.


EXAMPLE 5 ([11], Example 5.3). Let us consider the flat family X in A3 de-fined over the affine line T by the ideal (x, z) ∩ (y, z) ∩ (x − y, z − t x). Note that Xis the union of three lines X1, X2 and X3 passing through the origin P (and lying in a

plane if t = 0). Our aim is to calculate the degrees of the Segre classes of X diagonally

embedded in X × X .

The normal cone CX (X × X) = Spec K [x, y, z, u, v, w]/J can be calculatedby a computer, getting

J = (t2xy − z2, z(t x − z), z(t y − z), t yw + t zv − 2zw, t xw + t zu − 2zw,

t2xv + t2yu − 2zw,w(t2uv − tuw − tvw + w2)) .

Observe that J is a bigraded ideal with respect to the variables x, y, z and u, v, wrespectively. Let A be the ring of coordinates of X × X localized at (x, y, z, u, v, w)and I = (x − u, y − v, z − w)A the ideal of the diagonal in A. Then the bidegreesof J are the generalized Samuel multiplicities c(I, A) of I in A and, by a computercalculation, c(I, A) = (6, 3, 0, 0). By the previous proposition one gets the degrees ofthe Segre classes:

deg s0(X, X × X) = 0, deg s1(X, X × X) = 3 = deg X.

The same results hold if t = 0.

We observe that, since X is the union of three lines, using the bilinearity of the

intersection cycle v(X, X) (see for example [16], Section 2.1), we have v(X, X) =[X1] + [X2] + [X3] + 6[P] (which holds also in the case t = 0). From this it follows

immediately that c(I, A) = (6, 3, 0, 0) and hence one obtains the degrees of the Segreclasses as above.

3.8. Generalized Samuel multiplicities and Whitney stratifications

We recall the following definitions.

DEFINITION 3. Let X ⊆ Pn be a d-dimensional complex projective variety,and let Y ⊂ X be a non-singular subvariety. We say that the pair (Xreg,Y ) satisfiesthe Whitney conditions at a point x0 ∈ Y if for each sequence (xi ) of points of Xregand each sequence (yi ) of points of Y both converging to x0 and such that the limits

limxi→x0 Txi X and limxi ,yi→x0 xi yi exist in the Grassmannians G(d, n) and G(1, n)respectively, one has:

(a) limxi→x0

Txi X ⊃ Tx0Y ,

(b) limxi→x0

Txi X ⊃ limxi ,yi→x0

xi yi .

We remark that (b) implies (a).

DEFINITION 4. AWhitney stratification of X (d = dim X) is given by a filtra-

tion of X by closed subsets Fi

X = F0 ⊇ F1 ⊇ · · · ⊇ Fd+1 = ∅


such that

(i) Fi \ Fi+1 is either empty or is a non-singular quasi-projective variety of purecodimension i (the connected components of Fi \ Fi+1 are called the strata of thestratification);

(ii) whenever S j and Sk are connected components of Fi \ Fi+1 and Fl \ Fl+1 re-spectively, with S j ⊂ Sk, then the pair (Sk, S j ) satisfies the Whitney conditions(a) and (b).

DEFINITION 5 (Polar varieties). Let L(k) be an (n − d + k − 2)-dimensionallinear subspace of Pn, 1 ≤ k ≤ d = dim X. The k-th polar variety (or polar locus) of

X associated with L(k) is

P(L(k), X) := closure of {x ∈ Xreg | dim(Tx X ∩ L(k)) ≥ k − 1}.

For k = 0 we set P(L(0), X) := X.

If L(k) is generic, we write Pk(X) = P(L(k), X) since it is well known thatP(L(k), X) is empty or equidimensional of codimension k in X and its degree does notdepend on L(k). If

L(0) ⊂ L(1) ⊂ . . . ⊂ L(d)

is a generic flag, then we have

X = P0(X) ⊃ P1(X) ⊃ . . . ⊃ Pd(X) .

The polar varieties defined here are different from the polar varieties of Gaffney-

Gassler defined in Section 3.7.

Let x ∈ X . Teissier showed that the sequence of multiplicities

m0 = ex (P0(X)), . . . ,md−1 = ex (Pd−1(X))

does not depend upon the choice of the general flag. Moreover he proved the following

result.

THEOREM 12 (Teissier [45]). The pair (Xreg,Y ) satisfies the Whitney condi-tions in x0 if and only if the sequence of polar multiplicities

m0 = ey(X),m1 = ey(P1(X)), . . . ,md−1 = ey(Pd−1(X))

is locally constant in Y around x0.

DEFINITION 6 (The stratifying function g). Let X ⊆ Pn be a d-dimensionalcomplex projective variety and let x a point of X.

Let A := OX×X,(x,x) and let I be the diagonal ideal in A. We define

g(x) := e(GI (A)) =d∑

i=0

ci (I, A) .


Note that dim A = 2d, that cd+1 = · · · = c2d = 0 and that

(c0(I, A), c1(I, A), . . . , cd(I, A))

is a refinement of the multiplicity cd(I, A) = ex X = e(OX,x ) of X at x .

Figure 1: The g-stratification of the surface x4 + y4 = xyz.

THEOREM 13 ([5], Theorem 4.2). Let X ⊂ Pn be a (reduced) surface andx ∈ X be a closed point. Then

X j := {x ∈ X | g(x) ≥ j}, j = 0, 1, . . .

are closed subschemes of X or empty, and the connected components of

Sg( j) := g−1( j) = X j \ X j+1

are the strata of a Whitney stratification of X (the coarsest one if n = 3).

EXAMPLE 6. Consider the surface X in C3 (or in P3) defined by the equationx4 + y4 − xyz = 0, whose singular locus is the z-axis (see Figure 1). We want to

determine the coarsest Whitney stratification. Using [1] we obtain for the generalized

Samuel multiplicities (c2, c1, c0) and the polar multiplicities (m0,m1) (both orderedby codimension) the following values:

Locus (c2, c1, c0) g (m0,m1)

X \ Sing X (1, 0, 0) 1 (1, 0)

z-axis (2, 2, 0) 4 (2, 0)

origin (3, 6, 0) 9 (3, 4)

Hence the Whitney stratification is given by

surface ⊃ z-axis ⊃ origin.


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AMS Subject Classification: 13, 14.

Rudiger ACHILLES, Mirella MANARESI, Dipartimento di Matematica, Universita di Bologna, Piazza di

Porta S. Donato 5, I-40126 Bologna, ITALIA

e-mail: [email protected], [email protected]


Syzygy 2005

D.A. Buchsbaum

ALLA RICERCA DELLE RISOLUZIONI PERDUTE


Abstract. A brief indication of the role of resolutions in diverse parts of algebra.

1. Introduction

First, I want to say how glad I am to be able to participate in this celebration of Paolo’s

60th birthday (although he is quick to tell me that he hasn’t yet arrived at that venerable

age: we must all wait until 29 May for that event to occur).

What I want to illustrate in this talk are the various ways it has helped to look

for or at complexes that may be resolutions of significant modules. Another way to say

this is that we’re looking at resolutions or complexes with appropriate grade-sensitivity

(in the case of commutative algebra) or which facilitate calculation of Ext modules (in

the case of representation theory). First, let’s look at some of the more classical or

historical developments (from a personal point of view).

• The proof of the Hilbert Syzygy Theorem stimulated my interest in “resolu-tions.” In this case a rather simple-minded construction (the Koszul complex) yielded

a fairly powerful result. Hence the idea that a great deal of information can perhaps

reside in complexes.

• A conversation with Emil Artin at the very beginning of my career made meaware of the need for a homological characterization of regular local rings. This con-

versation stimulated further interest in resolutions, and Grobner’s book on Algebraic

Geometry [14] first led me to observe the fact that given an exact sequence

0→ A → B → C → 0,

the mapping cone of a map of a resolution of A into one of B, produces a resolution

of C . It took a little work to show that if both given resolutions were minimal, and the

map between them local, then so was the mapping cone. We’ll see more on that theme

later.

• Checking out the first step of the theorem:

pdRM + codimM = codim R,

involved the use of the minimal resolution of a module of projective dimension equal

to 1.

• Proving that gldim(R) ≥ edim(R) (embedding dimension) involved explicitconstruction of the Koszul complex as a direct summand of any resolution of the

residue field, k, of R.

373

374 D.A. Buchsbaum

So reasons for studying resolutions and special complexes abound; here are a

few areas in which I’ve resolutely dug in:

(a) Generalized Koszul Complexes

(b) Resolutions of determinantal ideals

(c) Resolutions of Weyl modules

We’ll take the above-mentioned items in turn.

2. Generalized Koszul Complexes

In [6], I introduced a family of complexes, the so-called Generalized Koszul Com-

plexes, and in [9] Rim and I developed many of their properties. We also defined a

more general notion of multiplicity; once we had a way of “resolving” an arbitrary

finitely generated module of finite length, it seemed reasonable to attach a multiplicity

to it and not just to a cyclic module of finite length.

The complexes defined in [6], however, were very fat; there were many redun-

dant terms, and they were generally unwieldy. Around ten years later, Eisenbud and I

[8] succeeded in writing down a corresponding family of “slimmed-down” complexes

which, in a particular case, coincided with the very elegant Eagon-Northcott complex

([13]). Although neither Eisenbud nor I was familiar with representation theory, it

turned out that our slimming-down involved certain modules which we learned a bit

later were the Weyl modules corresponding to “hook” partitions, namely, representa-

tions of a very special type.

Although there now were two classes of “generalized Koszul complexes,” no

attempt was made to directly connect them until two years ago (as far as I know),

when Boffi and I showed ([4]) that the fat complexes were homotopically equivalent

to the slim ones. This permits us, among other things, to carry a certain ‘homothety

homotopy’ that was used to prove the grade-sensitivity of the fat complexes, over to

the slim ones. One possible explanation for there having been no direct comparison of

these complexes way back in the 70’s is that nobody working on them knew enough

about representations to be able to make the comparison.

3. Resolutions of determinantal ideals

Shortly after Eisenbud and I had learned that we had used “hooks” in our complexes,

Eisenbud was approached by A. Lascoux, a student, at the time, of Verdier in Paris

(where Eisenbud was visiting for the year), who asked him to suggest some application

of Lascoux’ recently developed methods (involving combinatorics and representation

theory) to commutative algebra. Eisenbud pointed out that certain representation mod-

ules had emerged in our work on complexes, that it had been long recognized that the

terms in the resolutions of determinantal ideals should be representation modules, so

that perhaps Lascoux could make a connection in that context.

Alla ricerca delle risoluzioni perdute 375

Since the methods of Lascoux were tied to characteristic zero, that is the restric-

tion he continued to work under. That restriction was understandable: one part of his

method involved the use of Bott’s theorem on vanishing of homology of vector bun-

dles, which held only in characteristic zero, and another part of his technique involved

the fact that the linear group is completely reductive over the rationals. Nevertheless,

Lascoux succeeded in writing down explicitly the terms of the resolutions of the ideal,

Ip, generated by the p × p minors of the generic m × n matrix over a commutative

ring containing the field, Q, of rational numbers. He also wrote down the terms ofresolutions of Weyl and Schur modules in terms of their Jacobi-Trudi and Giambelli

matrices (in the Grothendieck ring of representations of the general linear group).

This work of Lascoux stimulated me to develop, with K. Akin and J. Weyman

([3]), a characteristic-free representation theory of the general linear group∗ with theidea of constructing the Lascoux type of resolution in general. It soon became clear

that, while it was possible to “universalize” the representation modules to arbitrary

commutative ground rings, the direct transcription of the Lascoux terms was not going

to yield the result we were looking for: a universal minimal resolution of the ideal,

Ip, generated by the p × p minors of the generic m × n matrix over the integers, Z.For one thing, even in the construction of the resolution of the ideal of submaximal

minors of the generic matrix ([2]), Akin, Weyman and I ran across so-called Z-formsof rational representations that had to be taken into account to ensure the acyclicity of

these resolutions.

To explain loosely what is meant here by Z-forms of rational representations,consider the rational representation of GL(F ⊗ Q), for some free abelian group, F ,over Z, denoted by S2(F ⊗Q), that is, the second symmetric power of F ⊗Q. Over Z,there are two distinct GL(F)-representations, D2(F) and S2(F), which yield equiv-alent representations, namely, S2(F ⊗ Q), when tensored with Q. Thus, D2(F) andS2(F) are called Z-forms of the rational representation, S2.

Another way to construct non-isomorphic Z-forms is the following:

Consider the short exact sequence

(1) 0→ Dk+2 → Dk+1 ⊗ D1 → K(k+1,1) → 0

where K(k+1,1) is the Weyl module associated to the hook partition (k + 1, 1). (We areleaving out the module F , as that is understood throughout.)

If we take an integer, t , and multiply Dk+2 by t , we get an induced exact se-quence and a commutative diagram:

0 → Dk+2 → Dk+1 ⊗ D1 → K(k+1,1) → 0

↓ t ↓ ↓0 → Dk+2 → E(t; k + 1, 1) → K(k+1,1) → 0,

where E(t; k + 1, 1) stands for the cofiber product of Dk+2 and Dk+1 ⊗ D1. Each of

these modules is a Z-form of Dk+1 ⊗ D1, but for t1 and t2, two such are isomorphic

∗It should be pointed out that both Carter and Lusztig ([12]) and Towber ([17]) had developed suchtheories a bit earlier. However, their constructions did not yield a category of representation modules large

enough to do what we had in mind.

376 D.A. Buchsbaum

if and only if t1 ≡ t2 mod k + 2 (see [1]). This says that Ext1A(K(k+1,1), Dk+2) ∼=Z/(k + 2), where A is the Schur algebra of appropriate degree. (For those famil-iar with the Yoneda definition of Ext, it should be pointed out that the generator of

Ext1A(K(k+1,1), Dk+2) is the exact sequence (1).)

But let’s return to the Lascoux resolutions of determinantal ideals and the ques-

tion of transporting them to universal resolutions over the integers. It was known that

the existence of such resolutions would imply that the Betti numbers of these ideals

are independent of the characteristic of the ground ring. However, since these ideals

don’t exhibit pathology exhibited by some other classes of ideals (such as the projec-

tive plane), for example they are perfect in every characteristic, it was believed that this

property must be true. But in [15] Hashimoto showed that the ideal of 2× 2 minors ofa generic 5 × 5 matrix has different Betti numbers depending on whether the ground

ring is the rationals or a field of characteristic three. Clearly the problem of resolving

determinantal ideals had to be rethought, and the arithmetic problems, such as Z-forms,had to be addressed.

4. Resolutions of Weyl modules

While the results recounted above were a bit disappointing, it was heartening to see

that, to produce the Hashimoto counterexample, it was necessary to have at hand the

tools to deal with representations in all characteristics. It was also fascinating to come

across the arithmetic problems presented by Z-forms, and to see their connection withour familiar Ext groups. It wasn’t long before it became apparent, at least to those of

us who are resolution-oriented, that it was time to confront the problem of resolving

classes of representation modules.

At first, Akin and I dealt with Schur modules, as Lascoux generally did, and

the reasonable type of “resolution” for these modules consisted of direct sums of ten-

sor products of exterior powers. For two-rowed skew-shapes, Akin and I had a com-

plete description of their resolutions (see [1]). However, when we started to consider

three-rowed shapes, the situation became so complicated that we weren’t sure that such

resolutions even existed; we had, after all, learned something from the determinantal

ideal situation. In order to at least reassure ourselves of their existence, we considered

Weyl modules and resolutions in terms of direct sums of tensor products of divided

powers which are, modulo certain constraints, projective modules over the Schur al-

gebra (the universal enveloping algebra for homogeneous polynomial representations

of the general linear group of given degree). Using induction on the “combinatorial

complexity” of the skew-shapes, and a mapping cone argument of the sort mentioned

in the Introduction, we were able to prove the existence of resolutions of Weyl modules

corresponding to all the shapes we were interested in. Then, by means of a “dualizing

functor,” we could prove the existence of the resolutions of Schur modules correspond-

ing to all these shapes. However, we were still far from explicitly describing these

resolutions and, after a few years of steady but terribly slow progress, we turned our

attention to other problems.

In the summer of 1990, G.-C. Rota and I met in Rome, and discovered that we


were both working on related problems. Consequently, we decided to work together,

bringing to bear on our common problems methods of combinatorics and homologi-

cal algebra. As a first test of the efficacy of this type of collaboration, we chose the

resolutions of Weyl modules.

It soon became clear that the maps that Akin, Weyman and I had called the

“Weyl maps” and the “box maps” in [3] were, once one translated the more algebraic

setup into letter-place terminology, certain compositions of so-called place polariza-

tions. And for place polarizations, there are classical identities (the Capelli identities)

which, with the more contemporary methods of multilinear algebra, could be general-

ized to higher powers to give:

Capelli Identities:

If k /= i

∂(r)i j ∂

(s)jk =

∑

α≥0

∂(s−α)jk ∂

(r−α)i j ∂

(α)ik

∂(s)jk ∂

(r)i j =

∑

α≥0

(−1)α∂(r−α)i j ∂

(s−α)jk ∂

(α)ik ,

where ∂(r)i j stands for the r

th divided power of the place polarization sending place j to

place i .

If i /= k and j /= l, then

∂(s)jk ∂

(r)il = ∂

(r)il ∂

(s)jk .

Using these Capelli identities, it was possible to simplify many of the very com-

plicated calculations that had proved so impenetrable earlier on. While there is still a

great deal of work left to do in connection with this problem of resolving Weyl mod-

ules, the terms of the resolutions have been completely written down (see [11]). The

maps of the resolutions are not yet explicitly described; their existence is guaranteed,

however, by the fact that for Weyl modules the resolutions being looked for are projec-

tive. The main trick in determining these resolutions goes back to the one mentioned

in the beginning of this article, namely, if

0→ A → B → C → 0,

is a short exact sequence, the mapping cone of a map of a resolution of A into one

of B, produces a resolution of C . In our current situation, the module, C , is a Weyl

module, and using a fundamental theorem on short exact sequences (again, see [11]),

we have Weyl modules A and B which are combinatorially less complex than C . Thus

an inductive argument (on degree of complexity) says that we have resolutions of A

and B of the desired type, and projectivity of the resolutions assures that there is a map

between them. The mapping cone, then, gives us our resolution of C . (The case of

Schur modules is taken care of by means of a duality functor between Weyl and Schur

modules introduced in [1].)

378 D.A. Buchsbaum

5. Concluding remarks

I do not know of any recent work on generalized Koszul complexes per se, beyond

that of [4]. There are still a number of mathematicians working on the generalized

multiplicity mentioned earlier, and one would hope that the simplifications provided

by use of the slimmer complexes might help in that regard.

On the question of resolving generic determinantal ideals, that is, ideals gen-

erated by minors of a given order of a generic m × n matrix, M. Hashimoto ([16])

has used the notion of tilting modules to prove the existence of resolutions consisting

of those kinds of modules. There are also other types of ideals which are of deter-

minantal type, such as Gorenstein ideals of codimension 3 (they are Pfaffians). It has

long been asked whether Gorenstein ideals of higher codimension can be described

similarly. I have no answer to that question, but I would offer a suggestion: make use

of representation theory if possible. By this I mean that, in the case of codimension

3, the Gorenstein ideals correspond to a particular “symmetry” in F ⊗ F , namely, to

%2(F) ⊂ F ⊗ F . In the case of higher codimension, there are more matrices to con-

sider, and one might try to rephrase these in terms of higher tensor products of copies of

free modules. These tensor products, in turn, decompose into “symmetries” (Schur or

Weyl modules), and the Gorenstein case may correspond to the correct choice of “sym-

metry.” This is admittedly vague, but if I could be more precise, I would be writing a

paper on that topic as well as this one. In any event, the point that I would like to make

is that representation theory has in large part been neglected as a tool in commutative

algebra (certainly not totally), and it would be interesting to see if a more aggressive

use of representation-theoretic methods would be productive.

On the question of resolving Weyl modules, the problems mentioned here have

been described in full detail in [11], as well as in a forthcoming book [5]. In the

latter, a thorough description of letter-place methods is given, along with polarizations

and other combinatorial techniques. In [1] and [7], connections are made between

resolutions of Weyl modules and intertwining numbers, a topic that is of interest in

modular representation theory. Here, too, one can see the application of homological

technique (Universal Coefficient Theorem) to certain areas of representation theory.

This, then, is the offering I make at this conference in honor of Paolo Valabrega.

If it has not convinced him of the ubiquity of resolutions, I hope that it does convince

him of my own personal resolve in relation to them.

References

[1] AKIN K. AND BUCHSBAUM D.A., Characteristic-free representation theory of the general linear

group. II. Homological considerations, Adv. in Math. 72 (2) (1988), 171–210.

[2] AKIN K., BUCHSBAUM D.A. AND WEYMAN J., Resolutions of determinantal ideals: the submaximal

minors, Adv. in Math. 39 (1) (1981), 1–30.

[3] AKIN K., BUCHSBAUM D.A. AND WEYMAN J., Schur functors and Schur complexes, Adv. in Math.

44 (3) (1982), 207–278.

[4] BOFFI G. AND BUCHSBAUM D.A., Homotopy equivalence of two families of complexes, Trans. Amer.

Math. Soc. 356 (8) (2004), 3077–3107.


[5] BOFFI G. AND BUCHSBAUM D.A., Threading homology through algebra: selected patterns, Oxford

University Press (Clarendon), Oxford 2006.

[6] BUCHSBAUM D.A., A generalized Koszul complex. I, Trans. Amer. Math. Soc. 111 (1964), 183–196.

[7] BUCHSBAUM D.A. AND FLORES DE CHELA D., Intertwining numbers; the three-rowed case, J.

Algebra 183 (1996), 605–635.

[8] BUCHSBAUM D.A. AND EISENBUD D., Generic free resolutions and a family of generically perfect

ideals, Adv. in Math. 18 (3) (1975), 245–301.

[9] BUCHSBAUM D.A. AND RIM D.S., A generalized Koszul complex. II. Depth and multiplicity, Trans.

Amer. Math. Soc. 111 (1964), 197–224.

[10] BUCHSBAUM D.A. AND RIM D.S., A generalized Koszul complex. III. A remark on generic acyclicity,

Proc. Amer. Math. Soc. 16 (1965), 555–558.

[11] BUCHSBAUM D.A. AND ROTA G.-C., Approaches to resolution of Weyl modules, Adv. in App. Math.

27 (1) (2001), 82–191.

[12] CARTER R.W. AND LUSZTIG G., On the modular representations of the general linear and symmetric

groups, Math. Z. 136 (1974), 193–242.

[13] EAGON J.A. AND NORTHCOTT D.G., Ideals defined by matrices and a certain complex associated

with them, Proc. Roy. Soc. Ser. A 269 (1962), 188–204.

[14] GROBNER W., Algebraische Geometrie I, II, Bibliographisches Institut AG, Mannheim 1968–1970.

[15] HASHIMOTO M., Determinantal ideals without minimal free resolutions, Nagoya Math. J. 118 (1990),

203–216.

[16] HASHIMOTO M., Auslander-Buchweitz approximations of equivariant modules, London Mathematical

Society Lecture Note Series 282, Cambridge University Press, Cambridge 2000.

[17] TOWBER J., Two new functors from modules to algebras, J. Algebra 47 (1977), 80–109.

AMS Subject Classification: 13D25, 18G10, 13.90, 20G05.

David A. BUCHSBAUM, Department of Mathematics, Brandeis University, South Street, Waltham, MA

02254 USA



Syzygy 2005

L. Chiantini

VECTOR BUNDLES, REFLEXIVE SHEAVES AND LOW

CODIMENSIONAL VARIETIES


Abstract. We give a short, but hopefully up–to–date account on the theory of vector bundles

and reflexive sheaves on projective spaces and hypersurfaces.

1. Introduction

The theory of vector bundles and reflexive sheaves has been developed rapidly from

the 70’s up to today, (see the fundamental papers [33], [35]) mainly because of its

connection with the geometry of low codimensional varieties. An important motivation

for the study of these objects came out from the conjecture ([34]) that any smooth

2-codimensional variety in Pr , r > 6, is complete intersection. Recently, a strong

impulse towards the theory followed the interest on the study of subvarieties of general

hypersurfaces.

In this note, we show a general approach which, at least for rank 2, seems to

us sufficient to justify the introduction and inspection of vector bundles and reflexive

sheaves, in the task of understanding the geometry of projective subvarieties.

The note is also a tribute to Paolo Valabrega, in the occasion of his 60th birthday.

Paolo was the advisor of my thesis and inspired all my early studies in Mathematics.

He addressed me to the theory of algebraic varieties, about 25 years ago. One of his

first intuition concerned the role that vector bundles play in the development of a point

of view for understanding the geometrical behaviour of low codimensional varieties.

I hope that writing down some fundamental aspects of the theory of rank 2

bundles and reflexive sheaves, will be useful for propagating a point of view in the

theory of projective varieties, that Paolo followed in his geometrical papers, in the last

years.

2. Subvarieties of projective spaces

Many concepts that will be displayed through the note, are in fact independent from the

characteristic of the ground field. Some of them (but it is not entirely clear which) do

not even need the presence of a ground field itself. Nevertheless, it is more comfortable

to work over an algebraically closed field of characteristic 0. So let us say that, through

the note, we only consider projective spaces and projective varieties defined over the

complex field C.

In the study of varieties of codimension one in the projective space Pr , one haslittle to deal with complicate relations between the local and global description.

381

382 L. Chiantini

Indeed hypersurfaces X are always described by a single equation, a homoge-

neous polynomial F in the ring C[x0, . . . , xr ], and properties of the hypersurface arestrictly connected with properties of F . For instance, deformations of X correspond to

deformations of the equation F (modulo the action of C∗).

In principle, if one wants to describe X in terms of 0-locus of functions, it

is necessary to deal with the local equations of X in the various affine pieces of Pr .Hence, there are transition functions which display X as the 0-locus of a section of a

line bundle, usually indicated as OPr (d), d being the degree of F . However, as theglobal sections of OPr (d) coincide with homogeneous polynomials of degree d, thenone could also forget this description and go on with the elementary algebraic point of

view.

As soon as one considers codimension 2, things become “dramatically” more

complicated and new methods cannot be avoided.

First of all, given X ⊂ Pr of codimension 2, by no means one can be sure of theexistence of two homogeneous polynomials F,G such that X coincides globally with

the locus where F = G = 0. Even locally, the existence of the two defining equations

is not guaranteed.

Varieties for which the two defining equations exist (on the whole projective

space) are (global) complete intersection. In general, a variety of codimension m

is a (global) complete intersection when there are r − m homogeneous polynomials

F1, . . . , Fr−m such that X coincides with the locus F1 = · · · = Fr−m = 0.

When the same condition is true locally, then X is locally complete intersection.

Things become a little easier when we restrict to study only smooth subvarieties

X ⊂ Pr , i.e. subvarieties without singular points. In this case, one can find a Zariskiopen cover of Pr such that, on any piece of the cover, X is defined by two polynomialequations. In other words, every smooth subvariety is locally complete intersection.

Turning back to the case of smooth varieties of codimension 2, one could expect

that the transition functions determined on the open cover by the equations of a smooth

subvariety X , could determine a rank 2 bundle E on Pr such that X coincides with the0-locus of a section of E . Unfortunately, this turns out to be false, in general. Indeed

the glueing conditions are not trivial on X .

When r = 2 and X is a discrete set of points, then the glueing conditions are

local on a finite set, so they behave friendly and one is always able to define a vector

bundle E out from some local equations of X .

On the other hand, when dim(X) ≥ 1, the glueing condition becomes non–

trivial. The resulting data define a line bundle on X , which is a twist of the canonical

bundle ωX , the line bundle defined by the differentials on X .

Serre’s construction ([64]) shows that one obtains a rank 2 bundle E on Pr

associated with X as soon as ωX is the restriction to X of some line bundle OPr (d).Varieties X satisfying this property are called subcanonical. The restriction of E to X

turns out to be isomorphic to the normal bundle of X : NX = TPr |X/TX .

For general varieties, the difference between ωX and the restriction of OPr (d)determines some singularity of E and E is no longer a vector bundle, but just a re-

Vector bundles and varieties 383

flexive sheaf, i.e. a torsion free sheaf such that the double dual E∗∗ is canonicallyisomorphic to E .

OPEN PROBLEM 1. It is still an open problem to determine a reasonable set of

conditions which extend the previous theory to varieties of codimension greater than 2.

In codimension 3, a theorem of M. Kreuzer ([41]) shows that any set of distinct

points in the 3-space can be realized as the 0-locus of a rank 3 bundle on P3. Thebundle, however, is in general far from being unique. It is not clear (and difficult)

to determine which invariants are allowed for a vector bundle of rank 3 with sections

vanishing exactly on a given set Z of points. Observe that any set of distinct points is

automatically subcanonical.

We do not know a precise description of curves in P4 which can be realized asthe 0-locus of sections of a rank 3 vector bundle. A theorem of Chang proves, however,

that not every smooth subcanonical curve in P4 shares this property ([10]).

For other varieties, observe that we have the following necessary condition:

(∗) All the Chern classes of the tangent sheaf (or the cotangent sheaf) of X are cut

on X by Chern classes of Pr .

In other words, the Chern classes of TX sit in the image of the restriction map

Chow(Pr )→Chow(X ).

We do not know to which extent the previous condition is also sufficient. It

coincides with subcanonicity for curves, thus it is sufficient for curves in P3 (and forcodimension 2) but it is not sufficient for curves in P4.

We guess that the condition could be sufficient only as soon as 2 dim(X)+1 ≥ r .

Other, more geometrical, generalizations of the notion of subcanonical varieties

could be tested. For instance, it would be interesting to know if the following could be

true:

(Q) A smooth surface X ⊂ P5 is the 0-locus of a section of a rank 3 vector bundle ifand only if there exists a (reflexive?) sheaf T of rank 2 on P5, whose restrictionto X is isomorphic to the tangent sheaf TX .

OPEN PROBLEM 2. Useless to say that even more complicate would be to de-

termine in general when a 2-codimensional subvariety X ⊂ Pr is the 0-locus of a rank2 torsion free sheaf, having singularities in a given dimension ≥ 0.

The subcanonical condition can be rephrased in terms of liaison.

Roughly speaking, we say that two subvarieties X , X ′ of codimension 2 are(directly) linked by the complete intersection Y of two hypersurfaces of degrees a, b,when Y = X ∪ X ′. We will write

X ∼a,b X ′.

Indeed one should consider a more refined definition, which takes care of the

384 L. Chiantini

case where X, X ′ have overlapping components. We refer to the paper [51] for a precisestatement.

It is classical that when X ∼a,b X ′, then the canonical class ωX of X is cut by

hypersurfaces of degree a+b−r−1 passing through X ′, outside the intersection X∩X ′.As a consequence, when X is subcanonical, with ωX = OX (e) (= OPr (e) ⊗ OX ) thenthere exists a hypersurface of degree a+b−r−1−e which exactly cuts X ∩ X ′ on X ′.Thus X ′ turns out to be described by the vanishing of three homogeneous polynomials,of degrees a, b, a + b − r − 1− e.

In general, one can see that we need r + 1 equations to describe point by point

a subvariety of codimension 2: namely, with two equations one describes a complete

intersection of codimension 2, containing X . The third equation restricts the intersec-

tion to X ∪ {a subvariety of codimension 3} and so on, increasing the codimension ofthe residue by 1 at every step, up to the reach of the empty set.

Thus subcanonical varieties are exactly those which are linked to subvarieties of

codimension 2 defined by 3 equations, which is the minimum allowed for non complete

intersection varieties.

For curves in P3, this remark exhausts the possible cases, for any curve in the3-dimensional projective space can be cut with 4 equations (see e.g. [66]).

What happens for higher dimensional object seems widely unknown, even for

the initial cases.

OPEN PROBLEM 3. In P4, a general (smooth) surface can be described by 5equations. Surfaces directly linked to subcanonical surface, can be cut by 3 equations.

The problem asks for a description of the geometry of surfaces which can be

described with 4 equations.

Even if their canonical class ω is not cut by hypersurfaces of P4, they have thefollowing property:

(∗∗) there exists a number u ∈ Z such that, in the Chow ring, the intersectionωX · (ωX ⊗ OX (u)) is 0.

The previous property should be also sufficient for a surface in P4 to be cut by4 equations.

In general the geometry (existence, properties, deformations, classification) of

this class of surfaces, which are, in a sense, a natural generalization of subcanonical

surfaces, is unknown.

The theory could be generalized in two ways: first considering varieties of codi-

mension 2 in Pr , r > 4, cut by 4 equations, then to varieties of codimension 2 cut by

any intermediate number of equations, between 4 and r .

Generalizations to subvarieties of higher codimension seem truly out of reach!


3. Subcanonical subvarieties

Let us restrict our attention to smooth subcanonical varieties X of codimension 2 in Pr .Also we assume in this section r ≥ 3, so that dim(X) ≥ 1.

We set e to be the number such that the canonical bundle ωX coincides with the

restriction OX (e) = OPr (e) ⊗ OX . e is also the index of speciality of X , i.e.

e = max{i : dim H0(ωX (−i)) > 0}.

Fixing e, we will also say that X is e-subcanonical.

In this setting, Serre’s correspondence shows that there exists a vector bundle E

on Pr and a section s ∈ H0(E) such that X coincides with the 0–locus of s. Identifyingany homogeneous summand of the Chow ring of Pr with Z, Serre’s correspondenceshows that the Chern classes of E are:

c1 = e + r + 1

c2 = deg(X).

Locally, s is defined by two polynomials in any set of an open cover of Pr . TheKoszul complexes of the two polynomials link globally to give the fundamental exact

sequence:

(1) 0→ %2E∗ = OPr (−c1) → E∗ = E(−c1) → IX → 0

where IX is the ideal sheaf of X .

From this description, it turns out that X is globally complete intersection of

hypersurfaces of degree a, b if and only if sequence (1) coincides with the (global)Koszul complex of two polynomials of degree a, b, i.e. when E splits:

PROPOSITION 1. In the previous setting, X is complete intersection of hyper-

surfaces of degree a, b if and only if the local Koszul complex globalizes, i.e. if andonly if E decomposes in a direct sum of line bundles E = OPr (a) ⊕ OPr (b).

Thus, the problem of detecting when a smooth subcanonical subvariety of codi-

mension 2 is complete intersection is rephrased, in the language of bundles, as a par-

ticular case of the general splitting problem:

SP: Determine conditions under which a vector bundle E of rank n decomposes into

a direct sum ⊕OPr (ai ).

An implicit answer to the splitting problem for rank 2 in P3 was determined in1942 by G. Gherardelli.

DEFINITION 1. We say that a variety X ⊂ Pr is t-normal if the surjection ofline bundles OPr (t) → OX (t) = OPr (t)⊗OX yields also a surjection of the spaces of

global sections H0(OPr (t)) → H0(OX (t)).

1-normal varieties are also called linearly normal varieties.

X is arithmetically normal if it is t-normal for all t .

386 L. Chiantini

In [28] Gherardelli proves the following result:

THEOREM 1. A curve C ⊂ P3 is complete intersection if and only if it is (a)subcanonical and (b) arithmetically normal.

A result of Strano ([65]) proves that a smooth 2-codimensional subvariety X ⊂Pr , r ≥ 4, is complete intersection if and only if its general hyperplane section is.

Thus, summing up these two results, we obtain:

COROLLARY 1. A smooth subvariety X ⊂ Pr , r ≥ 3, of codimension 2 is

complete intersection if and only if it is subcanonical and its general curvilinear section

is arithmetically normal.

To understand Gherardelli’s result for the splitting of rank 2 bundles, one needs

to consider the cohomology of E .

REMARK 1. The natural sequence

(2) 0→ IX (t) → OPr (t) → OX (t) → 0

shows that X is t-normal if and only if H1(IX (t)) = 0. From the cohomological

sequence associated to (1), one obtains

H1(IX (t)) = H1(E(−c1 + t)).

Thus:

A rank 2 bundle on P3 decomposes in a sum of line bundles if and only if

H1(E(i)) = 0 for all i .

The previous splitting criterion has been extended in 1962 by A. Horrocks, to

vector bundles of any rank in any projective space.

THEOREM 2 (Horrocks splitting criterion). A vector bundle E of rank n in Pr ,r > 1, decomposes in a sum of line bundles if and only if:

(3) Hm(E(i)) = 0 for all i and for all 1 ≤ m ≤r

2

We will refer to vector bundles satisfying condition 3 as arithmetically Cohen-

Macaulay vector bundles (aCM for short).

Let us just observe that Gherardelli’s theorem, Strano’s result and Kleiman’s

proof of the fact that for any vector bundle E , a general section of E(t) (see [39]), fort ; 0, vanishes in a smooth locus, indeed imply Horrocks criterion for rank 2 vector

bundles in Pr , r > 2.

Horrocks criterion has been improved in several ways. A contribute of Paolo

Valabrega to this theory is contained in the paper [18], where he proves that the van-

ishing of one single, well–determined cohomology group H1E(t0) is enough to causethe splitting of a rank 2 vector bundle in P3:


THEOREM 3 (Splitting criterion in P3). A vector bundle E of rank 2 in P3

decomposes in a sum of line bundles if and only if H1(E(t0)) = 0, where

t0 = −c1

2− 1 for even c1, or

t0 ∈{−c1 + 3

2,−

c1 + 1

2,−

c1 − 12

}for odd c1

Equivalently: a curve C ⊂ P3 is complete intersection if and only if it is e-subcanonicaland (t0)-normal, where

t0 =e

2+ 1 for even e, or

t0 ∈{e + 1

2,e + 3

2,e + 5

2

}for odd e.

Examples prove that this description is sharp.

REMARK 2. A brief account on the history of this result. The original proof

was based on the Gruson–Peskine speciality lemma (see [31]), which bounds the index

of speciality e.

As pointed out in [53] and [60], indeed, the use of the speciality lemma can be

avoided in the proof of the result.

The same theorem has been subsequently re–obtained by a direct examination

of the cohomology of vector bundles ([52]) or via the theory of the spectrum ([22]).

In the case of rank 2 bundles in Pr , r > 3, by using several times the hyperplane

sectional sequence:

(4) 0→ E(−1) → E → E ⊗ OH → 0

where H is a general hyperplane, then the splitting criterion of Theorem 3 yields:

THEOREM 4. A vector bundle E of rank 2 in Pr decomposes in a sum of line

bundles if and only if

H1(E(t0)) = H2(E(t0 − 1)) = · · · = Hi (E(t0 − i + 1)) = 0, i = 1, . . . , r − 2,

where

t0 = −c1

2− 1 for even c1, or

t0 = either −c1 + 3

2,−

c1 + 1

2,−

c1 − 12

for odd c1.

An important improvement of the Horrocks criterion has been obtained by Evans

and Griffith([29]):

388 L. Chiantini

THEOREM 5. A vector bundle E of rank n in Pr (r > 1) decomposes in a sum

of line bundles if and only if:

(5) Hm(E(i)) = 0 for all i and for all m = 1, . . . ,min{n − 1, r − 1}.

OPEN PROBLEM 4. For rank 2 bundles, the splitting criterion of Evans and

Griffith reads:

E splits if and only if H1(E(t)) = 0 for all t.

So we have two different conditions which imply the splitting of a rank 2 bundle: a

“flat” condition, by theorem 5:

H1(E(t)) = 0 for all t

and a “tower” condition, by Theorem 4

H1(E(t0)) = H2(E(t0 − 1)) = · · · = Hi (E(t0 − i + 1)) = 0, i = 1, . . . , r − 2

where t0 is a precise number, depending only on the first Chern class of E .

The “intersection” of these two conditions yields the following problem:

Q: Is H1(E(t0)) = 0 for some fixed t0 enough to guarantee the splitting of a rank 2

bundle in Pr , r > 3?

Indeed there are few known examples of non–splitting rank 2 bundles in P4, allof them essentially obtained from the Horrocks–Mumford bundle ([38]).

In all of these examples, H1E(t0) does not vanish.

It is not a consequence of Theorem 3, but still true, that a rank 2 bundle E on

P2 decomposes in a sum of line bundles if and only if a precise cohomology group

H1(E(t0)) vanishes. Here the choices of numbers t0, for which the statement holds,widens (see [4]).

Indeed, as observed in the introduction, the correspondence between subvari-

eties of codimension 2 in P2 (i.e. sets of points) and rank 2 vector bundles is weakerthan in higher dimensional spaces.

In P2, any non-empty set of points X has some non–vanishing cohomology

group H1(IX (t)), thus the notion of “arithmetically normal” becomes meaningless.

Also the notion of “subcanonical” variety is useless, for the theory of lines

bundles on such a finite X trivializes. One has to use instead the notion of Cayley–

Bacharach number.

e is a Cayley–Bacharach number for the finite set X ⊂ P2 if, for any pointP ∈ X , one has

H0(IX (e)) = H0(IX−{P}(e)).

In other words, there are no curves of degree e passing through X − {P} and missingP .


If e is a Cayley–Bacharach number for X , then one finds a rank 2 bundle E

on P2 and a section s of E which vanishes exactly on X . The Chern classes of E arec1 = e + 3, c2 = deg(X).

From this description, one easily understands that there are infinitely many bun-

dles associated to X . As Cayley–Bacharach numbers associated to X may change, we

find bundles associated to X with different Chern classes.

Sets of points in P2 which are complete intersection, are characterized by theCayley–Bacharach theorem:

THEOREM 6 (Cayley–Bacharach). A set of point X ⊂ P2 of degree d = ab

is complete intersection of curves of degree a, b if and only if a + b − 3 is a Cayley–

Bacharach number for X.

Among the many rank 2 bundles associated to a complete intersection X ⊂ P2,there are for sure indecomposable bundles. Nevertheless, it is possible to determine

cohomological criteria for the splitting of rank 2 bundles on P2.

Namely, the possible sequences which give the dimension of the cohomology

groups H1(E(t)), as t varies, are classified in [4]. It turns out that we have a crite-rion for detecting, from the cohomological sequence dim H1(E(t)), the existence ofa global section of some twist E(t) which vanishes on a complete intersection set ofpoints.

We refer to the paper [4] for details.

OPEN PROBLEM 5. If E is a rank 2 vector bundle on P2, then the possiblestructures of the cohomology ⊕H1(E(t)), as vector spaces over C, are known, asexplained above. Nevertheless ⊕H1(E(t)) is also a module over the polynomial ring.It is not known which module structures are allowed for this object.

Actually we only know a factorial characterization of the cohomology module

⊕H1(E(t)), via its minimal resolution (see [21] for details).

What we have in mind is a sort of results similar to the one of Ellia and Fiorentini

([24]):

THEOREM 7. The only indecomposable rank 2 bundles in P3 whose cohomol-ogy module⊕H1(E(t)) has trivial multiplication are associated to pairs of skew lines.

Indeed Theorem 3 is a step in this direction. See also [37] for some bounds on

the dimension of H1(E(t)) for rank 2 bundles in P3.

OPEN PROBLEM 6. A similar description for the structure of⊕H1(E(t)), whenE is a rank 2 vector bundle on Pr , r > 2, has been exploited quite partially. E.g.

Decker’s result [21] also works in P3.

In fact, in Pr , r > 2, we do not know a complete description of all the possible

functions dim H1(E(t)) for a vector bundle E of rank 2. We just have partial results.E.g. Buraggina proved in [8] that the support of the function is connected. Also Chang

390 L. Chiantini

gave a conjecture on the non–vanishing of cohomology groups for rank 2 bundles (see

[9], [27], [20]).

When one considers codimension 2 subvarieties in Pr , with r ≥ 6, then a new

phenomenon occurs: namely the Lefschetz principle implies that all smooth subvari-

eties are subcanonical (see e.g. [32]). So any smooth subvariety of codimension 2 is

associated with a rank 2 vector bundle.

Classical algebraic geometers knew very well the difficulties in constructing

smooth subvarieties of low codimension. For codimension 2, there are no examples of

such varieties in Pr as soon as r ≥ 5 (but we should mention that such a subvariety

exists in P5, in characteristic 2, see [45]).

For r ≥ 7, in [34] Hartshorne conjectured that:

CONJECTURE 1. All smooth subvarieties of codimension 2 in Pr , r ≥ 7, are

complete intersection.

Equivalently:

All rank 2 bundles in Pr , r ≥ 7, decompose in a sum of line bundles.

It is far beyond the scopes of this note to give an account of the state of the art

in the study of Hartshorne’s conjecture. Enough to say that a proof of this conjecture

would bound the study of rank 2 bundles over projective spaces Pr to the case of lowr .

Turning back to the case of curves in P3, let us mention another suggestionwhich comes from the splitting Theorem 3.

OPEN PROBLEM 7. Let C be a smooth curve in P3 of degree d and genus g.

Then the index of speciality e of C ranges between 0 and2g−2d.

e is maximal for subcanonical curves. Assume for a while that e is even, e = 2a.

Then Theorem 3 yields that, for any degree, a subcanonical curve C is arithmetically

normal if and only if H1(IC (a + 1)) = 0.

The other extremal case is e = 0. Curves for which this condition holds are

called non–special. A theorem of Mumford ([47]) shows that, for d > 2g − 1, then

a non special curve in P3 is arithmetically normal if and only if it is linearly normal,i.e. H1(IC (1)) = 0. Observe that in this case also a = e/2 is zero, thus Mumford’scondition can be written as the previous one: H1(IC (a + 1)) = 0.

The question is: can this pattern be extended to the intermediate cases?

Q: is there some (serious) function,(g) such that for smooth curves in P3 with evenindex of speciality e = 2a, and for d > ,(g), arithmetically normal is equivalentto the vanishing of just the unique cohomology group H1(IC (a + 1))?

It could happen that the number a + 1 should be replaced by some more com-

plicate expression involving e, d, g.

Of course, a similar problem can be stated for curves with odd index of special-


ity, or for surfaces in P4.

The previous problem suggests to exploit how the behaviour of subcanonical

subvarieties is mirrored by general subvarieties of codimension 2. We consider here

mainly the case of curves in P3 and surfaces in P4.

Fix a smooth curve C in P3 and call e its index of speciality. Then, glueingtogether the various local equations of C , one ends up with a sheaf F which is not

locally free, unless C is subcanonical. F is just a reflexive sheaf of rank 2, a rank 2

“vector bundle with singularities”. The singularities of F are the points where it is not

locally free, i.e. the points at which the fiber of F jumps in dimension. These points

belong to C , and their number depend on the construction of the glueing. Namely,

taking any value ε ≤ e, one gets a reflexive sheaf F with Chern classes c1 = ε + 4,

c2 = deg(C) and c3 = 2g− 2− εd (where g is the genus of the curve), such that thereis a section s ∈ H0(F) which vanishes exactly along C .

As above, by taking the glueing of the local Koszul complexes, one gets an exact

sequence:

(6) 0→ OP3 → F → IX (c1) → 0

In this situation, F is singular exactly at c3 = 2g − 2− εd points. Thus F is arank 2 bundle exactly when c3 = 0.

A completely analogue construction works also in P4 and P5. It could work alsoin higher dimensional projective spaces, but due to Barth–Larsen Theorem [32], every

smooth subvariety of codimension 2 is subcanonical in such spaces, thus the reflexive

sheaf theory here is useless.

One may arise, for reflexive sheaves, the same splitting problem that exists on

rank 2 bundles: i.e., find cohomological conditions under which the reflexive sheaf F

(is a vector bundle and) decomposes into a sum of line bundles.

It turns out easily that F decomposes if and only if, for all t , H1(F(t)) =H2(F(t)) = 0. Observe that, since F is not locally free, the H2 sequence could be, in

principle, completely different from the H1 sequence.

The cohomological criterion has been refined by M. Roggero, for curves in P3

(see [54], and [60] for the extension to higher dimensional spaces):

THEOREM 8 (Roggero’s criterion). A reflexive sheaf F of rank 2 on P3 withfirst Chern classes c1 ∈ {−1, 0} is a vector bundle and decomposes into a sum of line

bundles, if and only if H2(F(t)) = 0 for some number t in the range [−3+c1,−3−c1].

Observe that the condition, via Serre duality, coincides with the one of Theorem

3 for vector bundles.

OPEN PROBLEM 8. Is there some analogue of Evans-Griffith criterion of The-

orem 5 for rank 2 reflexive sheaf on higher dimensional spaces?

Can this be intersected with Roggero’s criterion?

392 L. Chiantini

OPEN PROBLEM 9. Let us mention that Strano’s criterion for the splitting of a

rank 2 reflexive sheaf extends indeed to any rank. Is this true also for other splitting

criteria?

The starting point of our discussion works verbatim, if we consider subvarieties

of codimension 2 of some variety Y with good geometrical properties.

Namely, for instance, if we consider a subvariety X ⊂ Y of codimension 2 in a

smooth complete intersection variety Y , then Serre’s correspondence also works: if X

is subcanonical, then one can find a rank 2 bundle E on Y , with a section s vanishing

along X . The exact sequence (1) now reads:

(7) 0→ OY (−c1) → E∗ = E(−c1) → IX,Y → 0.

where again we use integers to signify the tensor product by line bundles on Y belong-

ing to the image of the restriction map:

ρ : Pic(Pr ) → Pic(Y )

which, in our cases, will usually be a surjection.

We have:

PROPOSITION 2. In the previous setting, E decomposes in a direct sum of line

bundles E = L ⊕ M if and only if X is complete intersection of the 0–loci of a section

of L and a section of M.

It turns out that the analogy between the theory of rank 2 bundles on Pr andon Y , stops soon when we consider splitting criteria, even for very simple complete

intersection Y .

EXAMPLE 1. Let Y be a smooth quadric in P4. Then there are lines X inside Y .X cannot be complete intersection of two surfaces in Y . Indeed the map ρ : Pic(Pr ) →Pic(Y ) surjects, thus any surface in Y arises by intersecting Y with an hypersurface inP4, thus any complete intersection curve in Y has even degree.

On the other hand, X is subcanonical, so we have a rank 2 bundle E on Y ,

associated with X , which cannot decompose. As X is arithmetically normal in P4,then the exact sequence 0 → OP4(−2) = IY → IX → IX,Y → 0 shows that

H1(IX,Y (t)) = 0 for all t . By the exact sequence (7), this in turn implies H1(E(t)) =0 for all t .

Hence Horrocks splitting criterion does not work on a general quadric in P4.

One can prove (see e.g. [50]) that the rank 2 bundle associated with a line X is

essentially the unique counterexample to the Horrocks splitting criterion, on a quadric

hypersurface Y of P4.

The situation for hypersurfaces of P4 has been clarified by C. Madonna, whofound in [42] that Horrocks splitting criterion is valid except for bundles whose first


Chern class fits in a well determined range. Indeed Madonna went over proving an

extension of the splitting criterion of Theorem 3:

THEOREM 9 (Madonna’s criterion). Let Y ⊂ P4 be a smooth hypersurface ofdegree d. Let E be a rank 2 vector bundle on Y . Set:

(8) b := max{i : H0(E(−i)) /= 0}

and assume

c1 − 2b ≤ −d + 2 or c1 − 2b ≥ d.

Then the following are equivalent:

(i) E decomposes in a sum of line bundles E = OY (a) ⊕ OY (b);

(i i) H1(E(t)) = 0 for all t .

Observe that Madonna’s range −d + 2 < c1 − 2b < d is empty when d = 1,

i.e. when Y = P3. Also it reduces to a singleton when d = 2, corresponding to the

case of the rank 2 bundle associated with a line.

Madonna’s range for c1 is indeed sharp. On one hand, if Y contains a line,

then the associated rank 2 bundle satisfies and contradicts Horrocks splitting criterion.

On the other hand, one find examples of rank 2 bundles with c1 = d − 1, when Y

corresponds to the pfaffian of a skew-symmetric matrix of linear forms (see [2] for

details).

OPEN PROBLEM 10. Madonna’s criterion has been extended to a wider class

of threefolds, as complete intersection threefolds or regular subcanonical threefolds.

See [42] for details.

However, an extension to more complicate classes of threefolds (for which the

Serre correspondence could present some failures) is unknown.

OPEN PROBLEM 11. Using Madonna’s criterion or direct constructions, one

could try to classify all rank 2 vector bundles on smooth threefolds X ⊂ P4, whichdoes not decompose and nevertheless satisfy H1(E(t)) = 0 for all t .

This has been done for deg(X) = 2 ([50]), deg(X) = 3 ([1]) and deg(X) = 4

([43]). For deg(X) = 5, in [15] one finds a description of all the bundles with the

previous property, which could exist on a smooth quintic. The effective existence is

not known (but see [40] for some partial results).

In general, on a smooth hypersurface of degree d > 5, it is not known, for a

fixed c1 in Madonna’s range, which values one has for the second Chern class of an

indecomposable aCM rank 2 bundle.

The situation becomes only a little easier when one looks at general hypersur-

faces X of P4. The situation does not change for deg(X) ≤ 5, while for deg(X) = 6,

i.e. for canonical hypersurfaces, we have:

394 L. Chiantini

THEOREM 10. On a general hypersurface X of degree 6 in P4, the Horrockssplitting criterion holds for rank 2 bundles. Namely, a rank 2 bundle E decomposes in

a sum of line bundles if and only if H1(E(t)) = 0 for all t .

As a consequence, Gherardelli’s criterion holds: a smooth curve in X is com-

plete intersection of X and two other hypersurfaces if and only if it is subcanonical and

arithmetically normal.

Proof. See [16].

OPEN PROBLEM 12. Although one can hope to play some induction on the

degree, by now we do not know whether Horrocks splitting criterion is valid for rank 2

bundles on a smooth hypersurface of degree d > 6 in P4.

REMARK 3. For rank 2 bundles on a general hypersurface of Pr , r > 4, the

splitting criterion works, except for quadrics in P5.

This has been proved for hypersurfaces of degree d = 3, 4, 5, 6 in P5 in [17] andextended to any degree in a recent paper by Mohan Kumar, Rao and Ravindra ([46]).

The extension to higher dimensional projective spaces follows easily by Strano’s crite-

rion.

The case d = 2, r > 5 is easy. In fact, by Madonna’s criterion, Horrocks split-

ting theorem would fail only when the quadric contains a linear space of codimension

2, which is not the case for general quadrics.

OPEN PROBLEM 13. Extend the previous theory to other classes of threefolds,

as complete intersections, subcanonical threefolds, etc.

OPEN PROBLEM 14. Is there some analogue of Evans-Griffith criterion of The-

orem 5 for rank 2 bundles on general hypersurfaces of Pr , r > 3?

OPEN PROBLEM 15. Some particular smooth sextic threefold X in P4 doesnot satisfies Horrocks splitting criterion. E.g. if X contains a line, then this line is

associated with a rank 2 bundle which is indecomposable and aCM.

The situation is similar to the Noether–Lefschetz principle for surfaces of degree

d > 3 in P3: we know that, on a general surface, all line bundles are restrictions of linebundles on the projective space. However, for some particular surface, new line bun-

dles may arise. Surfaces which do not satisfy the Noether–Lefschetz principle, fill up

some subsets of the parametrizing space P(H0(OP3(d))), which are called Noether–Lefschetz loci. The study of these Noether–Lefschetz loci has several applications in

Algebraic Geometry.

Analogously, one may ask to describe the locus in P(H0(OP4(6))) which para-metrizes hypersurfaces of degree 6 in P4 bearing some indecomposable aCM rank 2

bundle.

OPEN PROBLEM 16. For the case of surfaces in P3, Madonna’s criterion forthe splitting of rank 2 vector bundles also works: Horrocks criterion is valide outside a


narrow range for the invariant b of Theorem 9.

It would be interesting to know a description of all indecomposable aCM rank

2 bundles on a general surface of degree d of P3.

For d = 2 this is more or less well known. The case d = 3 has been solved by

Faenzi ([25]), while the case d = 4 is essentially well known. For d = 5, a forthcoming

paper ([14]) will describe the situation.

The problem is open for higher values of d. Giving a precise description of

all possible shapes of the function t <→ dim(H1(E(t))), for vector bundles on smoothsurfaces of P3, seems a task. However, using a lemma on the Hilbert function of setsof points on surfaces ([13]), in a forthcoming paper [5] we are ready to give serious

restriction for the possible shapes of the function.

At the end of this section, we would like to point out that, as in the case of P3,non–subcanonical curves C lying in smooth hypersurfaces X ⊂ P4 are associated withrank 2 reflexive sheaves F on X .

M. Valenzano (see [67]) extended to this situation a mix between Madonna’s

and Roggero’s criteria:

THEOREM 11. Let F be a reflexive sheaf of rank 2 on a smooth hypersurface

X ⊂ P4 of degree d. Set b := max{i : H0(E(−i)) /= 0} and assume

c1 − 2b ≤ −d + 2 or c1 − 2b ≥ d.

Then F is a vector bundle and decomposes in a sum of line bundles F = OX (a) ⊕OX (b) if and only if

H2(F

(d − c1 − 6

2

))= 0 for even d − c1;

H2(F

(d − c1 − 7

2

))= 0 for odd d − c1.

OPEN PROBLEM 17. It is clearly not an easy task, but it could be worthy of

some effort, the attempt to apply the methods used for rank 2 bundles, to obtain in-

formation on the Chern classes of reflexive sheaves on smooth hypersurfaces of P4 (oreven more general threefolds).

This could be a way to answer some interesting question about curves contained

in general threefolds of P4. E.g. questions like the following ones are still unanswered:

• Q1: Is it true that every smooth, connected arithmetically normal curve on a

general smooth threefold of degree d > 5 in P4 has degree multiple of d?

• Q2: Is the statement true for subcanonical curves?

See [30] and [68] for a discussion on the subject.

396 L. Chiantini

Finally, in order to construct examples of subcanonical curves in smooth three-

folds, one can try to consider double structures on curves, following the pattern intro-

duced by Ferrand in P3 (see [26]). In [6] one finds some generalization to multiplestructures. In [44] one can find a systematic study of such structures.

Let us point out that a generalization of Ferrand construction has been per-

formed in Valabrega’s paper [18], §3. The construction yields examples of subcanoni-

cal curves and vector bundles whose cohomological properties can be controlled rather

deeply. Indeed, several pathological examples of rank 2 bundles on P3 can be obtainedin this way (see e.g. [11]).

OPEN PROBLEM 18. Reproduce the multiple structure construction of [18] for

curves and varieties in more general spaces, and exploit which kind of bundles one may

obtain.

4. Families of varieties and vector bundles

The study of deformations of vector bundles and reflexive sheaves has a natural interest

in projective Algebraic Geometry. Let us set the problem from the very beginning.

Any hypersurface X of the projective space Pr is defined by a single homoge-neous equation F ∈ C[x0, . . . , xr ]. In order to deform X we need just to deform the

homogeneous polynomial F . The set of all hypersurfaces that can be obtained with

a deformation of X coincides thus the set of hypersurfaces defined by homogeneous

equations of the same degree of F . They are parametrized by the points of a projective

space

|OPr (d)| = P(H0(OPr (d)))

where d is the degree of F . Hence, at this level, any hypersurface is described by its

degree plus a point in a projective space.

If we replace Pr with a (even smooth, irreducible) projective variety V , thensome of the previous facts fail.

A hypersurface X ⊂ V , i.e. a subvariety of codimension 1 in V , is no longer

globally defined, modulo the homogeneous ideal of V , by a unique equation. Never-

theless, as V is smooth, X is defined by a unique equation in a suitable Zariski open

neighbourhood of any point of V . These equations may vary as the Zariski open set

varies. They match in the overlapping of the neighbourhoods, modulo well defined

transition functions. Thus X determines a line bundle LX over V and a global section

sX ∈ H0(LX ), with the property that X is exactly the locus where sX vanishes. This istrue not only set-theoretically, but in a refined algebraic sense, i.e. scheme-theoretically.

In order to deform X in V , we may clearly take a deformation of the section

sX ∈ H0(LX ). So the hypersurfaces of X associated to global sections of LX are the“first–level” deformations of X .

However, depending on the geometry of the variety V , it is possible that a defor-

mation of the transition functions of LX could determine also a deformation of the line

bundle, which carries a deformation of the global section s. We get thus a family of line


bundles and for any G in the family and any non–zero global section t ∈ H0(G), thelocus defined by t = 0 is also a hypersurface obtained with a deformation of X . Line

bundles G obtained from a deformation of LX are indeed of the type LX + ε, whereε is a line bundle which lies in a variety (Pic0(V )) which parametrizes line bundlesnumerically equivalent to 0.

Thus, deformations of X are parametrized by points of a space which is a pro-

jective fibration over a subvariety of Pic0(V ).

Pic0(V ) is trivial when X is rational, or a hypersurface of dimension≥ 2. On the

other hand, when for instance V is a non–rational curve, then Pic0(V ) has a complicategeometrical structure, exploited in the Brill–Noether theory.

When we consider subvarieties X of codimension 2 in a projective space (or,

even worse, in a subvariety Y ) then the structure becomes suddenly much more difficult

to describe.

First of all, we do not have a nice organization of such objects in terms of

sections of some vector bundle.

An organization of this type would follow, from Serre’s construction, only in

the case of subcanonical varieties. For the general case, one should refer to reflexive

sheaves, which however are not uniquely determined, even in their numeric features,

by the subvariety X .

Restricting our attention to subcanonical varieties X , nevertheless one has a

notion of “linear” deformation: fix a rank 2 bundle associated to X and a section s ∈H0(E) which vanishes along X and take a deformation of s inside P(H0(E)). Thisis, roughly speaking, the concept equivalent to a deformation of local equations for

subvarieties of codimension 1. Besides that, there are the deformations that one can

reach by moving the pair (bundle, global section).

This sounds as a very reasonable theory, but unfortunately the space parametriz-

ing vector bundles of rank 2, even on a projective space, may have a very intricate

structure, whose understanding is far from being complete. The situation worsens if

we replace Pr with some variety Y .

OPEN PROBLEM 19. A systematic study of the deformations of a subcanonical

variety, from this point of view, has been scarcely carried on. Even a collection of

fundamental results, as the space parametrizing subcanonical deformations, is sparse

in a set of papers which, at least occasionally, deal with the subject.

See e.g. [63], [12], [27] for examples.

A similar problem of course could be introduced for non–subcanonical varieties,

just replacing the rank 2 bundle with a rank 2 reflexive sheaf.

In any event, it seems natural to believe that the organization of deformations

of 2-codimensional varieties in terms of linear deformations plus deformations of the

bundle (or the reflexive sheaf) could lead to some new point of view in the theory.

One more reason to look at the vector bundle in order to understand subvarieties

of codimension 2 is given by the notion of “twist”.

398 L. Chiantini

Namely, if X is the 0–locus of a section of a rank 2 vector bundle E , then 0–loci

of sections of a twist E(n) = E ⊗OPr (n) yields subvarieties which, for many features(cohomology, deformations, etc.) are similar to X . Thus one can change X with some

new subvariety X ′ in order to obtain geometrical information, play induction, and soon.

The general non–sense rule, in this game of passing from 0–loci of sections of

E to 0–loci of sections of E(n) is:

• n ; 0 =⇒ geometrically simpler, arithmetically more complicate

• n ? 0 =⇒ geometrically more complicate, arithmetically simpler.

Beware that starting with a smooth subcanonical variety X and taking sections

of E(n), n ? 0, one may easily obtain varieties with very bad singularities (even

non–reduced ones!).

However we have the following Bertini-type theorem ([39]):

THEOREM 12. Assume that E is generated by global sections, i.e. there exists

a surjective map O tPr

→ E for some t. Then a general global section of E has smooth

zero locus.

Thus, by replacing E with E(n), n ; 0 (when possible), one may always

assume that the zero locus of a general section of E is smooth.

Observe that the Castelnuovo–Mumford criterion [48], §14, provides a cohomo-

logical procedure to determine when a vector bundle (actually, any torsion free sheaf)

is generated by global sections:

THEOREM 13 (Castelnuovo-Mumford criterion). If Hi (E(−i)) = 0 for all i >0, then E is generated by global sections.

This also suggests the following (probably non–sense, in any case very hard):

OPEN PROBLEM 20. Determine necessary and sufficient conditions on E, nsuch that a general section of E(n) has smooth zero locus.

OPEN PROBLEM 21. Provide conditions on E under which, for all n, H0(E(n))/= 0 implies that a general section of E(n) has smooth zero locus.

OPEN PROBLEM 22. Assume that E has a global section whose zero locus is

smooth. Assume that E(1) has sections which vanish in codimension 2. May oneconclude that a general section of E(1) has smooth zero locus?

A basic question in this theory is

Q): Given a vector bundle E, for which n one has H0(E(n)) /= 0;

and a similar question clearly arises for reflexive sheaves.


Consider the case rank(E) = 2 and E has a section whose zero locus is X . Via

the usual exact sequence

0→ OPr → E → IX (c1) → 0

the question can be rephrased as follows:

assume we are given a subcanonical variety X of codimension 2. For which values d

are there hypersurfaces of degree d which contain X?

Notice that the rephrasement, however, is not perfectly coherent. Namely

min{d : H0(IX (d)) /= 0} = min{d : H0(E(d − c1)) /= 0}

unless E is strongly normalized, i.e. unless dim H0(E) = 1. Indeed when E is nor-

malized, min{d : H0(E(d − c1)) /= 0} = c1 but X lies in no hypersurfaces of degree

c1.

OPEN PROBLEM 23. Find “serious” geometrical conditions on a subcanonical

variety X of codimension 2 which determine whether X is the zero locus of a section

of a strongly normalized rank 2 bundle.

We know indeed that subcanonical varieties X, X ′ which are zero loci of sec-tions of E, E(n) respectively, for the same rank 2 bundle E , are doubly linked. Namelythere exists a variety Y which is directly linked both to X and X ′. With this respect,zero loci of sections of strongly normalized rank 2 bundles often have minimal degree

in their equivalence class with respect to the relation: X ∼ Y ⇐⇒ X is doubly linked

to Y .

Going back to the main argument of this section, let us notice that the question

about the minimal degree d of a hypersurface which contains a given variety X is a non

trivial one. In general, X is always contained in a hypersurface of degree deg(X) ([18],§2). However this bound, which turns out to be sharp for linear subvarieties, is rather

coarse.

In this setting, the main result is due to Hartshorne, who proved in [36]:

THEOREM 14. Let F be a rank 2 reflexive sheaf in P3, with second Chern classc2 ≥ 0 and c1 ∈ {0,−1}. Then H0F(n) /= 0 provided that n >

√3c2 + 1 − 2 if

c1 = 0, or n >√3c2 + (1/4) − (3/2) if c1 = −1.

This result, although partial, it is indeed sharp in its range.

Improvement are possible if one knows something on the cohomology of E .

For instance, Paolo Valabrega proved in [18] the following result:

THEOREM 15. Let E be a rank 2 bundle in P3. Assume H0(E) /= 0 and

H1(E(c1)) = 0. Then c2 ≥ c1√c1 + 2c1 − 2

√c1.

Notice that, for the case of P2, almost any answer comes out from our knowl-edge of the shape of the function t <→ dim H1(E(t)) (see e.g. [4]).

400 L. Chiantini

Beware that Kleiman’s Theorem 12 does not work for reflexive sheaves. Namely

there are reflexive sheaves F such that no sections of F(n) vanish in a smooth locus, forall n. Reflexive sheaves having sections with smooth zero locus are called curvilinear.

For general reflexive sheaves, Paolo Valabrega and M. Roggero proved the fol-

lowing bound ([59]), which works for any Chern classes:

THEOREM 16. Let F be a reflexive sheaf in P3 having a non zero global section.Then H0(F(t)) /= 0 for some t ≤ c1 + 2

√3c2 + 1+ 3c1/4.

The same authors extended these methods to produce similar bounds for sub-

canonical surfaces and rank 2 bundles in P4.

OPEN PROBLEM 24. Repeat this analysis for rank 2 vector bundles and re-

flexive sheaves on threefolds different from P3, as general hypersurfaces, completeintersections, etc.

When the vector bundle or reflexive sheaf E on P3 is strongly normalized, thenthe minimal degree of a surface containing the zero locus of a section of E evidently

does no longer represent the minimal level of twist such that E has sections. Instead it

is connected with the level of a second section of E :

DEFINITION 2. Let F be a reflexive sheaf of rank 2 on P3. Call α the minimal

integer such that H0(F(α)) /= 0. Then for all n ≥ α the space H0(F(n)) containsall the products of a section of H0(F(α)) by forms of degree n − α. These productsvanishes along a surface.

Define β as the minimal integer such that

dim H0(F(β)) > dim H0(OP3(n − α)).

In other words, β is the first level of twist such that F(β) has a section whichdoes not depend algebraically from a minimal section.

REMARK 4. If dim H0(F) > dim H0(F(−1)) = 0 and C is the zero locus of

a (minimal) section of E , then c1 + β is the minimal degree of a surface containing C .

If Y is the smooth irreducible zero locus of a section of F(n), for n ; 0, then Y

is contained in a complete intersection curve of minimal type (n+α + c1, n+β + c1).

Roggero proved in [53] that the general section of F(n) vanishes along a curveexactly when either n = α or n ≥ β.

Thus the knowledge of this second section level has relevant applications in the

theory of vector bundles and curves.

In particular, it allows to start with the procedure of liaison, which produces

exactly the chance of reading properties of a given curve in some other, hopefully

simpler, variety.

In this setting, the best known results has been found by Paolo Valabrega, Rog-


gero and Nollet. In a series of papers ([55], [57], [58], [59]) culminating with [49],

they prove the following:

THEOREM 17. Let F be a rank 2 reflexive sheaf, with c1 = 0,−1. Considerthe numbers α, β defined in 2. Let T0 be the largest real root of the polynomial:

T 3 − (6c2 + 6αc1 + 6α2 + 1)T + 3(2α + c1)(c2 + c1α + α2).

Then β ≤ T0 − α − c1 − 1.

The same paper contains applications of this result to determine the minimal

degree of a surface containing a given curve, where the word curve in this setting

means any generically complete intersection subscheme of pure dimension 1 (possibly

reducible and non reduced). This generalization of the question about the geometry

of a curve to such non–trivial varieties produced recently a huge amount of literature,

which we do not record here.

OPEN PROBLEM 25. A similar analysis for rank two bundles and reflexive

sheaves on surfaces and threefolds which are not projective spaces is only at a very

initial stage.

OPEN PROBLEM 26. Only few ideas are known about the extensions of these

theories to codimension 3 or higher, e.g. to rank 3 bundles.

Finally, let us point out quickly another huge field of researches in this theory.

As we said at the beginning of the section, besides the study of curves or vari-

eties arising from the section of one fixed vector bundle, one may deform the bundle

itself, just as one does in the study of linear systems.

Classification spaces (Moduli spaces) for vector bundles are extensively studied

in the literature, and we will not even try to present here an overview of the theory,

neither a short patch of it.

Let us simply recall a long–standing open problem, which is strictly related to

the above discussion on the cohomological properties of vector bundles.

It is well known that, in the border of most Moduli spaces of vector bundles,

one finds also non–locally free sheaves, which are, in general, just torsion free.

For instance, starting with the decomposable bundle OP3(2) ⊕ OP3(2), whosegeneral sections correspond to an elliptic quartic, complete intersection of two quadrics

Q1, Q2, and moving the quadrics to acquire a common plane, there is a limit curve Cwhich splits in the union of an elliptic plane curve and a line. Accordingly, the bundle

OP3(2) ⊕ OP3(2) is the general element of a family whose special member E0, whichcorresponds to C , is not locally free. Notice that, in any event, H1(E0(n)) = 0 for all

n.

The cohomology groups are semi–continous in flat families, and there are ex-

amples of flat deformations in which the general elements have some vanishing co-

homology groups which do not vanish for the special element (see e.g. [12]). Nev-

402 L. Chiantini

ertheless there is no known example of splitting bundles of rank 2 which degener-

ate to non–splitting ones. After Horrocks criterion, this means that we do not know

whether H1(E(n)) = 0 for all n and for all general elements in the family, implies that

H1(E0(n)) = 0 for any special element which is locally free.

As sections of E(n) vanish on smooth varieties, for n ; 0, the problem can be

exactly rephrased as follows:

• Are there flat families of smooth curves in P3 whose general member is completeintersection, while some special member is not?

The problem, although apparently quite basic, seems very hard. Only partial

answers are known.

E.g. one gets a negative answer by imposing that all the members of the family

belong to surfaces of small degree ([19]).

On the other hand, Mohan Kumar proved in [45] that there are examples of this

type in positive characteristic.

For an overview of the setting of the theory, one can read the paper [19].

The interesting feature of this problem resides in the (quite unusual) fact that

no precise feeling seems available, through the mathematical community, on what the

answer should be.

This justifies the absence of conjectures.

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AMS Subject Classification: 14F05.

Luca CHIANTINI, Dipartimento di Scienze Matematiche e Informatiche R. Magari, Pian Dei Mantellini

44, 53100 Siena, ITALIA



Syzygy 2005

E. Gover

MAXIMAL POINCARE SERIES AND BOUNDS FOR BETTI

NUMBERS


Abstract. We give a new proof of an existing theorem concerning maximal growth rates of

Betti numbers. Other results that use Poincare series formulas to give upper bounds for the

growth of Betti numbers of finitely generated modules over local rings are then surveyed.

1. Introduction

Consider Notherian local rings (A,m, k) and (B, n, k), a finitely generated

B-module M , and a local morphism f : A → B inducing an isomorphism of residue

class fields A/m ∼= B/n ∼= k and through which B and M become finite A-modules.

Denote the nth Betti numbers of B and M over A by bAn (B) and bAn (M) respectively,and the nth Betti number of M over B by bBn (M). The corresponding Poincare seriesare then the formal power series PA(B), PA(M), and PB(M) whose coefficients arethose Betti numbers.

With the additional technical assumption that TorA(k, B) is a k-vector spacewhen considered as a B-module (which is automatically satisfied when f is surjec-

tive), an inequality was established by the author and P. Salmon in [9] that relates the

coefficients of the Poincare series:

(1M ) PB(M) APA(M)

1− t (PA(B) − 1).

(The symbol A signifies that the inequality is among corresponding coefficients of thepower series in the variable t on either side.) In terms of Betti numbers, (1M ) can beexpressed as

bBn (M) + bBn−1(M) ≤n∑

j=1

bBn− j (M)bAj−1(B) + bAn (M)

for n ≥ 1, and bB0 (M) ≤ bA0 (M). The relationship is proved by considering the changeof rings spectral sequence with

E2p,q = TorBp

(TorAq (k, B),M

)⇒pTorAp+q(k,M)

and then counting dimensions of appropriate k-vector subspaces. The inequality is the

natural extension to finitely generated modules of one originally stated by Serre [19] in

407

408 E. Gover

the special case where M = k. In its general form, (1M ) gives implicit bounds for thegrowth rates of the Betti numbers of M .

Questions concerning the growth rates of Betti numbers for classes of modules

over local rings have been considered by several investigators, especially Avramov. In

[2], he proved that the Betti numbers of arbitrary finitely generated modules over local

rings have at most strong exponential growth. The purpose of the present article is to

reprove that and a related result directly from inequality (1M ). Then, with an upperbound for the growth of Betti numbers in place, a review will be given of growth rates

at the extremes, namely, for modules over complete intersections and Golod rings.

2. Exponential bounds for the growth rates of Betti numbers

We may assume, using the Cohen structure theorem, that all local rings considered are

homomorphic images of regular local rings. That is, we may take the n-adic completionB of local ring B to be isomorphic to A/b where (A,m, k) is regular and b ⊆ m2. AsB is flat over B, the homological data including the Betti numbers and Poincare series

will be the same for B and B, and the former may be replaced by the latter. The

condition b ⊆ m2, which can always be achieved, assures that A and B have the sameembedding dimension: e.dim A = dimk(m/m2) = dimk(n/n2) = e.dimB. Bounds for

the growth rates of Betti numbers now follow using (1M ).

THEOREM 1. The Betti numbers of a finitely generated module M over any lo-

cal ring B have at most termwise exponential growth; that is, for a given module, there

exists a constant α > 1 (which depends on M and B) and a sequence of positive inte-

gers {cn} such that bBn (M) ≤ cn for each n, and cn ≥ α cn−1 for all n ≥ dimk(m/m2).The bounding sequence {cn} also exhibits strong exponential growth; namely, there ex-ist constants 1 < β ≤ γ such that βn < cn < γ n for all n ≥ 2 dimk(m/m2), andtherefore bBn (M) < γ n for those n.

Proof. Both assertions follow from inequality (1M ) applied to the surjective naturalmap f : A → A/b = B where A is regular and b ⊆ m2. Define {cn} by setting

(2)

∞∑

n=0

cntn =

PA(M)

1− t (PA(B) − 1).

Then, by (1M ), bBn (M) ≤ cn for each n. The next part of the argument parallels

Peeva [18, Proposition 5] but in a slightly different context. As A is regular, both

the A-free minimal resolutions of M and B, X A(M) and X A(B), are finite, so theEuler characteristics of these resolutions vanish implying t = −1 is a root of bothPA(M) = 0 and PA(B) = 0. Thus, we may write

PA(M) = (1+ t)pM (t) and PA(B) = (1+ t)pB(t)

where pM (t) and pB(t) are polynomials in t of degree < ε = dimk(m/m2).

Maximal Poincare series and bounds for Betti numbers 409

When B is not a hypersurface ring (so that bA1 (B) ≥ 2 and a2 = bA1 (B) −bA0 (B) /= 0), denominator 1− t (PA(B)− 1) = 1+ t − t P A(B) = (1+ t)(1− tpB(t))has the form

1− t (PA(B) − 1) = (1+ t)(1− t − a2t2 − · · · − ar t

r )

where an =∑n

i=1(−1)i+1bAn−i (B) for 2 ≤ n ≤ r ≤ ε and a2, ar /= 0. It follows that

equation (2) can be rewritten as

∞∑

n=0

cntn =

pM (t)

1− t − a2t2 − · · · − ar tr

from which for each n ≥ ε,

(3) cn = cn−1 + a2cn−2 + · · · + ar cn−r .

Localizing at (0) and considering the Euler characteristic of the appropriate exact se-quence, dimk

(SyzAn (B)(0)

)− an = 0 (where SyzAn (B) is the nth syzygy module of

X A(B)), thereby showing for each n ≤ r that an is a positive integer. Meanwhile,

starting from the bottom using (2), cn = bAn (M) ≥ 1 for n = 0, 1 (assuming M is not

A-free) and after simplification,

cn =n∑

j=2

cn− j bAj−1(B) + bAn (M)

for n ≥ 2, which shows that cn ≥ 1 for all n ≤ ε. When combined with (3), this showsthat cn > cn−1 for all n ≥ ε, provided B is not a hypersurface ring.

Once the cn strictly increase for n ≥ ε − 1, set α = min{cεcε−1

,cε+1cε

, . . . ,c2ε−1c2ε−2

}.

Then α > 1, and for all n with ε ≤ n ≤ 2ε − 1, it follows that cn ≥ αcn−1. Supposethat for some n ≥ 2ε it has already been shown that c j ≥ αc j−1 for all ε ≤ j ≤ n− 1.Then,

cn = cn−1 + a2cn−2 + · · · + ar cn−r ≥ αcn−2 + a2αcn−3 + · · · + arαcn−r−1 = αcn−1.

The inequality cn ≥ αcn−1 now holds for all n ≥ ε by induction, which establishesthe first assertion of the theorem for all except hypersurface rings. For those rings,∑∞

n=0 cntn = pM (t)/(1 − t), in which case cn = cn−1 for n ≥ ε. Thus, the Betti

numbers of finitely generated modules over hypersurface rings are bounded and hence

eventually constant by a result of Eisenbud [4].

Strong exponential bounds for the Betti numbers over non-hypersurface rings

can now be found using (3). To find an upper bound γ > 1 such that cn < γ n forn ; 0, start by considering (3) with n = 2ε:

c2ε = c2ε−1 + a2c2ε−2 + · · · + ar c2ε−r .

Set λ = 1 +∑r

j=2 a j . Then, because c2ε−1 > · · · > c2ε−r , it follows that c2ε <

λc2ε−1. Repeating the calculation using n = 2ε + 1 gives c2ε+1 < λc2ε < λ2c2ε−1.

410 E. Gover

By induction, c2ε+s < λs+1c2ε−1 for any s ≥ 0. Moreover, c2ε−1 > 1 so it may be

rewritten as c2ε−1 = µ2ε−1 with µ > 1. Thus, c2ε+s < λs+1µ2ε−1 for any s ≥ 0.

If γ = max{λ, µ}, then c2ε+s < γ s+1γ 2ε−1 = γ 2ε+s with γ > 1. In other words,

cn < γ n for all n ≥ 2ε.

Finding a lower bound β for which βn < cn is even easier and so that argument

will be skipped. Note that the lower bound holds for n ≥ ε + 1. Taken together, when

n ≥ 2ε, both bounds apply and βn < cn < γ n .

REMARK 1. In [2], Avramov proved the second part of the theorem in a dif-

ferent way by applying a theorem of Fatou, which says that when∑

n≥0 cntn repre-

sents a rational function and the coefficients cn are eventually non-negative and non-

decreasing, then the cn exhibit strong exponential growth if and only if the radius of

convergence of the power series is less than 1. For the rational function considered in

equation (2) of Theorem 1, the denominator

1− t (PA(B) − 1) = 1+ t − t P A(B) = 1− bA1 (B)t2 − · · · − bAr (B)tr+1,

is a polynomial with a single positive root α < 1 while the numerator has no positive

root. Hence, the radius of converge is < 1 and so the growth of the cn is strong

exponential by Fatou’s result.

3. Cases where Poincare series inequality (1M) becomes an equality

We recall some relevant terminology:

DEFINITION 1. A local morphism f : (A,m, k) → (B, n, k) for which inequal-ity (1M ) is an equality when M = k and for which nH(X A(k) ⊗A B) = 0 holds,

is called a Golod homomorphism. In the second condition, X A(k) denotes a mini-mal A-free resolution of k over A and H signifies reduced homology. This condition

holds automatically when f is surjective. When A is regular and the natural map

f : A → B = A/b is a Golod homomorphism with b ⊆ m2, the ring B is called a

Golod ring. If M is a finitely generated B-module for which (1M ) is an equality, M is

called an f -Golod module.

There are various characterizations of Golod rings, Golod homomorphisms, and

f -Golod modules. For example, results of Levin from [14], [15], [16] give conditions

equivalent to the defining ones and are summarized in the next two theorems.

THEOREM 2. The following are equivalent:

(1) f is a Golod homomorphism; that is, P B(k) =PA(k)

1− t (PA(B) − 1)and

nH(X A(k) ⊗A B) = 0.

(2) The induced maps TorA(k, k) → TorB(k, k) and TorA(k, n) → TorB(k, n) areinjective.


(3) The induced map TorA(k, k) → TorB(k, k) is injective and TorA(k, B) has trivialMassey products.

(4) There exists a minimal set of generators V for H(X A(k) ⊗A B) and a trivialMassey operation γ on V with Im(γ ) ⊂ n(X A(k) ⊗A B).

THEOREM 3. The following are equivalent:

(1) The B-module M is f -Golod; that is, P B(M) =PA(M)

1− t (PA(B) − 1).

(2) f is a Golod homomorphism and the induced map TorA(k,M) → TorB(k,M) isinjective.

(3) TorA(k,M) → TorB(k,M) and TorA(k,SyzB1 (M)) → TorB(k,SyzB1 (M)) areboth injective.

(4) For every i > 0, SyzBi (M) is f -Golod.

Some comments clarify these equivalences. Regarding the first three conditions

in each theorem, recall that inequality (1M ) is obtained from the change of rings spec-tral sequence with

E2p,q(M, f ) = TorBp (TorAq (k, B),M) ⇒

pTorAp+q(k,M).

Meanwhile, the induced maps mentioned in the second and third conditions of each

theorem are specific cases of

TM ( f ) : TorA(k,M) ∼= H(X ⊗A M) ∼= H((X ⊗A B) ⊗B M)H( f⊗1M )−−−−−−→ TorB(k,M)

where X = X A(k) and Y = Y B(k) are minimal resolutions of k over A and B respec-tively, and f : X ⊗A B → Y is a lifting of 1k . It turns out that the edge maps of the

spectral sequence together with the filtration that results from its convergence lead to

a factorization of TM ( f ) that forces injectivity of TM ( f ) when inequality (1M ) is anequality. Conversely, if Tk( f ) and TM ( f ) are injective, inequality (1M ) becomes anequality. This suggests how the equivalence (1)⇔ (2) in each theorem can be obtained.

For properties of Massey products see, for example, [11]. Massey products are

used in essentially two ways in Theorem 2. First, Golod’s original idea [6], updated by

Gulliksen [10] can be expressed in the following way:

LEMMA 1. If a connected DG A-algebra % has trivial Massey products and is

free of finite type as an A-module, and if T (L) denotes the tensor algebra of the gradedA-module L where L0 = 0 and Ln is a free A-module of rank= dimk(Hn−1(%)⊗A k),then the differential on % can be extended to a differential on Y = % ⊗A T (L) so thatY becomes an A-free resolution of k. If, moreover, ∂% ⊂ m% and a trivial Massey

operation γ can be chosen for a minimal set of generators of H(%) so that Im(γ ) ⊂m%, then Y is a minimal resolution of k.

412 E. Gover

Basically, trivial Massey products guarantee existence of a trivial Massey oper-

ator γ , which is used to extend the differential to Y = %⊗A T (L) in such a way that Yis acyclic and so becomes an A-free algebra resolution of k. The additional conditions

∂% ⊂ m% and Im(γ ) ⊂ m% ensure that this resolution is minimal. When the lemma

is applied in the context of the local morphism f described in Theorem 2, the result is:

COROLLARY 1. Set X = X A(k). If there exists a trivial Massey ope-

ration γ defined on a minimal set of generators for H(X ⊗A B) with

Im(γ ) ⊂ n(X ⊗A B), then TorA(k, k) → TorB(k, k) is injective and inequality (1M )with M = k is an equality.

Proof. Let L be the free, graded B-module with L0 = 0 and rank |Ln| = |Hn−1(X⊗A

B)| for n ≥ 1. By the lemma, (X ⊗A B) ⊗B T (L) is a minimal B-free resolution of kand therefore TorB(k, k) ∼= (X⊗A k)⊗B T (L). At the same time, TorA(k, k) ∼= X⊗A k

and T0(L) = B, so the natural map X ⊗A k → (X ⊗A k) ⊗B T (L) is injective, thusdemonstrating the first assertion.

The second assertion also follows from TorB(k, k) ∼= (X ⊗A k) ⊗B T (L). Interms of Hilbert series, this identification becomes

PB(k) = H(X ⊗A k)H(T (L)) = PA(k)/(1−H(L)),

whileH(L) = t (PA(B) − 1) holds by virtue of the construction of L .

The corollary establishes (4)⇒ (1) and parts of (4)⇒ (2) and (3) of Theorem 2.

The remaining implications (2)⇒ (3)⇒ (4) of that theorem follow by interpreting the

kernels of the maps TM ( f ) in terms matric Massey products, which are generalizationsof Massey products that are due to May [17]. Details of how such products are used

in Theorems 2 and 3 can be found in [15]. Details of the remaining implications of

Theorem 3 can be found in [16].

Many Golod homomorphisms are surjective with f : A → A/b = B where

b ⊆ m2. The ideal b is called a Golod ideal. Examples of Golod ideals include thefollowing:

• b = 0. (In other words, the identity 1A : A → A is a Golod homomorphism.)

• b = x I where x ∈ m is regular and either ideal I is proper or x ∈ m2. [15]

• b =∑n

i=1 yi Ii where I1 ⊆ · · · ⊆ In ⊆ m and yi ∈ m are such that y1 is regular,

and for j > 1, y j is regular on A/∑

i< j yi Ii . (These ideals include as11 · · · asnn

where the ai are generated by disjoint parts of a regular sequence.) [9]

• b = mt for all t ; 0. [14]

• b = I (r, s) = ideal of s × s minors of an r × s matrix (r ≥ s ≥ 2) with entriesin m and depth A I (r, s) = r − s + 1. [1]

• b = (0 : m) where A is a 0-dimensional Gorenstein ring of e.dim > 1. [15]


Another example of a Golod homomorphism [10] is the natural homomorphism

A → A(M) where A(M) is the trivial extension of A by an A-module M .

In each of the first five examples, if the ring A is regular, A/b is a Golod ring.In the fourth example, A/mn is a Golod ring for any n ≥ 2 when A is regular as a

consequence of either the third or the fifth example, or of [7].

By Theorem 2, Golod rings are also rings where TorA(k, B) has trivial Masseyproducts. TorA(k, B) can be computed as H(X A(k) ⊗A B) or as H(k ⊗A X A(B)).Resolving the first argument with A regular, X A(k) is just the Koszul complex K A de-

fined by a minimal set of generators for maximal ideal m of A. The condition b ⊆ m2

ensures that X A(k) ⊗A B = K A ⊗A B = K B , which is the corresponding Koszul

complex over B defined by a minimal set of generators for maximal ideal n. A Golodring is therefore one where K B has trivial Massey products. Note that if B is a com-

plete intersection, H(K B) is an exterior algebra, so K B has trivial Massey products

only if H2(KB) = 0 and K B has just one generator. Thus, hypersurface rings are both

complete intersections and Golod rings. In all other cases, complete intersections be-

have quite differently from Golod rings. That difference will be discussed in the next

section.

When TorA(k, B) is computed using a resolution of the second argument, B isa Golod ring provided that k ⊗A X

A(B) has trivial Massey products where X A(B) isa minimal resolution. In fact, it suffices that k ⊗A W

A(B) has trivial Massey productswhere W A(B) is any free resolution of B over A. Rings of the form B = A/I (r, s)where A is regular and depth A I (r, s) = r − s+ 1 were first proved to be Golod by thisapproach using a generalized Koszul complex to resolve B [7]. Later, when an algebra

structure was given for the Eagon-Northcott complex [21], which serves as the minimal

resolution for such rings, triviality of Massey products was noted there as well.

Examples of f -Golod modules include:

• M = 0 over any A for any f .

• M = any finitely generated module over any A and f = 1A : A → A.

• M = any finitely generated module over any A and f : A → A/(x), where x isa non-zero-divisor and x ∈ m(annM). [20]

• M = k and M = SyzBi (k) for all i when f is a Golod homomorphism, by

Theorem 3. [16]

• M = any finitely generated module over A/mn for n ; 0 such thatmn−1M = 0

and f : A → A/mn . [15]

• M = any finitely generated B-module such that (0 : n)M = 0 in the case where

f : A → B is a strong Golod homomorphism, that is, where there is a chain map

H(X A(k)⊗A B) → X A(k)⊗A B inducing an isomorphism on homology where

H(X A(k) ⊗A B) is regarded as a complex with trivial differential. (See [15].)Such modules include, in particular, the proper ideals of B. It is shown in [15]

that strong Golod implies Golod. The maps f : A → A/mn for n ; 0 of the

414 E. Gover

preceding example are strong Golod homomorphisms (and the f -Golod modules

of that example are precisely the ones with (0 : n)M = 0.) Other examples of

strong Golod homomorphisms are trivial extensions A → A(M) where M is

a k-vector space, and A → A/nr where t ∈ n − n2 is such that t2 = 0 and

nr = tnr−1.

• N = any finitely generated A-module regarded as an A(M)-module and f : A →A(M) is the natural map to the trivial extension of A by the finitely generatedA-module M , [16]. In this case, N becomes an A(M)-module via the projectionmap A(M) → A.

For a Golod ring, the proof of Theorem 1 applies directly to the sequence of

Betti numbers of the residue field and shows that this sequence exhibits strong expo-

nential growth. Put another way, over a Golod ring B the residue field k is f -Golod,

where f : A → A/b = B is the map defining B as a Golod ring. By Theorem 3, all

SyzBn (k) are f -Golod as well. The sequences of Betti numbers of these syzygy modulestherefore also exhibit strong exponential growth, again using the proof of Theorem 1.

Even if A is not regular, fn : A → A/mn is a strong Golod homomorphism

for sufficiently large n (determined using the Artin-Rees lemma). If M is a finitely

generated (A/mn)-module, then mn−1SyzA/mn

1 (M) = 0, so this syzygy module and

therefore also all higher syzygies are fn-Golod. This was used in [8] to show for rings

of Krull dimension d ≥ 2 that the Betti numbers bA/mn

i (M) of non-free, finitely gen-erated (A/mn)-modules strictly increase for all i ≥ 2. The result was later improved

by Lescot [12] to show for d ≥ 2, any n ≥ 2, and i ≥ 1 that the sequence {bA/mn

i (M)}strictly increases.

4. Rings and modules whose Betti numbers grow at the extremes of the permissi-

ble range

Two questions: When are the bounds given in Theorem 1 for the growth rates of Betti

numbers achieved? What are the possible growth rates for sequences of Betti numbers

of finitely generated modules over a particular ring?

For Golod rings, the story is found in [2] and [18], which use results from [5]

and [13] to obtain:

THEOREM 4. For a finitely generated module over a Golod ring A, precisely

one of the following situations must occur:

(1) pdAM < ∞.

(2) A is a hypersurface ring, pdAM = ∞, and the Betti numbers of M are eventually

constant and nonzero after depth A− depthM + 1. Moreover, SyzAn (M) (and thereforethe minimal resolution) is periodic of period 2 after that point.

(3) A is not a hypersurface ring, pdAM = ∞, and the Betti sequence of M has strong

exponential growth. In more detail, the Betti sequences of such modules strictly in-

crease after degree 2ε − 1 where ε is the embedding dimension of A, exhibit termwise


exponential growth after degree 2ε, and have strong exponential growth after degree3ε.

The situation for complete intersections is in marked contrast, except for the

shared case of a hypersurface ring. A complete intersection is a local ring B whose

completion is of the form B ∼= A/a with A regular and a generated by an A-regularsequence x1, . . . , xr ⊂ m2. If the embedding dimension of B is e.dimB = ε andthe codimension is r = ε − depth B, then B is characterized [22] by the form of the

Poincare series associated with its residue field, k, which is:

PB(k) =(1+ t)ε

(1− t2)r .

A hypersurface ring is a complete intersection B with r = 1. In this case,

X A(B) is just 0 → Ax1−→ A → A/(x1) → 0, making 1 − t (PA(B) − 1) = 1 − t2.

The denominator thus has the right form for a Golod ring. The numerator also has the

right form: (1+ t)ε = PA(k) because with A regular, the Koszul complex K A defined

from A → A/m is a minimal A-free resolution of k and also an exterior algebra on

ε generators. Therefore, in this special case, B is both a complete intersection and aGolod ring. In all other cases, complete intersections and Golod rings behave quite

differently from each other.

The form of PB(k) for complete intersections implies that the Betti numbersbBn (k) for all n ≥ 0 are given by a polynomial in n of degree r − 1:

bBn (k) =ε−r∑

i=0

(ε − r

i

)(n + r − 1− i

r − 1

).

Hence, the growth of these Betti numbers is polynomial of degree r−1. It turns out thatfor finitely generated modules over complete intersections, all growth of Betti numbers

is polynomial of specific degrees. The description, due to Avramov, Gasharov and

Peeva [3], utilizes Avramov’s notion of complexity.

DEFINITION 2. The complexity of M over A, denoted cxAM, is d if d−1 is thesmallest degree of a polynomial in n that bounds bAn (M) from above. The zero poly-

nomial is assigned degree −1, and cxAM = 0 means the zero polynomial eventually

bounds the Betti numbers; in other words, pdAM < ∞.

Complexity cxA M = 1 means that a constant bounds the Betti numbers. Com-

plexity cxA M = ∞ signifies that no bounding polynomial exists. Polynomial growth

that is of degree r−1, for example, the growth of the Betti numbers bBn (k), is expressedby saying that cxB(k) = r = e.dimB − depthB. For all other finitely generated M ,

formula (8.5) of [3] gives

(4M ) PB(M) =pM (t)

(1− t)d(1+ t)e

416 E. Gover

with pM (t) ∈ Z[t] such that pM (±1) /= 0. (Note that integer e should not be con-

fused with e.dimB = ε.) In this setting, Theorems (8.1) and (8.6) of [3] become thefollowing:

THEOREM 5. If B is a complete intersection and M is a finitely generated B-

module whose Poincare series is given by (4M ), then cxB M = d ≤ e.dimB−depthB,pM (1) > 0, and one of the following cases holds:

(0) d = 0 : e < 0 or e = 0 with pM (−1) > 0; also

deg pM (t) = depthB − depth BM + e,

(1) d = 1 : e ≤ 0 and deg pM (t) = depthB − depth BM + e.

(2) d ≥ 2 : e < d − 1 or e = d − 1 with pM (1) > |pM (−1)|.

For case (2) with n ; 0, the Betti numbers bBn (M) are given by polynomials b+(n)when n is even and b−(n) when n is odd, with b+(t), b−(t) ∈ Q[t] and

b±(t) =b

2e(d − 1)!td−1 +

c±

2d(d − 2)!td−2 + lower order terms

with integers b, c±, and e such that either 0 ≤ e ≤ d − 2, c+ = c−, and b > 0,

or else e = d − 1 and b > |c+ − c−|. In particular, both difference polynomialsb±(t + 1) − b∓(t) have degree d − 2 and positive leading coefficients.

REMARKS 1. In case (0) of the theorem where d = 0, pdBM < ∞ and PB(M)is a polynomial. In case (1) of the theorem, d = 1 means the Betti numbers are

bounded. Eisenbud [4] showed that bounded sequences of Betti numbers for finitely

generated modules over complete intersections are eventually constant and that they

become periodic of period 2 after at most (dim B) + 1 steps.

In case (2), d ≥ 2 and limn→∞ bBn (M)/nd−1 = the common leading coefficient

of the polynomials b±(t). When it comes to limn→∞(bBn (M)−bBn−1(M))/nd−2, how-ever, the situation is different—the limit exists when c+ = c− but does not exist whenthey are unequal.

To illustrate this, consider A = k[[X,Y ]] and B = A/(X3,Y 3). Thus, B isa complete intersection with ε = 2 and codimension r = 2, so by the Tate formula

shown above, PB(k) = (1 + t)2/(1 − t2)2 = 1/(1 − t)2, which implies bBn (B/n) =bBn (k) = n+1 for each n ≥ 0. On the other hand, bBn (B/n2) = 3

2n+1 for even n ≥ 0,

and bBn (B/n2) = 32n + 3

2for odd n ≥ 1. (See [2].) These Betti numbers give different

values, namely 1 and 2, for (even) − (odd) as opposed to (odd) − (even), so the limit

of the differences does not exist.

Thus, there is a complete description of the asymptotic behavior of Betti num-

bers of finitely generate modules over complete intersections and a different description

over Golod rings. For complete intersections, growth is polynomially bounded with

detail added using the notion of complexity; for Golod rings that are not hypersurface


rings, all growth is exponential. There are other rings over which all growth is either

polynomial or exponential and where some of each occurs.

THEOREM 6. [2], [3]. Let B a local ring that satisfies one of the conditions:

(1) B is one link from a complete intersection;

(2) B is two links from a complete intersection and B is Gorenstein;

(3) e.dimB − depthB ≤ 3;

(4) e.dimB − depthB = 4 and B is Gorenstein.

If M is a finite B-module whose Poincare series has radius of convergence

ρ ≥ 1, then cxB M = d ≤ e.dimB − depthB and there exist polynomials !1 and !2,

each of degree d − 2 with positive leading coefficients, such that for n ; 0,

!1(n) ≤ bBn (M) − bBn−1(M) ≤ !2(n).

In particular, {bBn (M)} is eventually either constant or eventually strictly increasingand bounded by polynomial growth.

If M is a finite B-module whose Poincare series has radius of convergence

ρ < 1, then, {bBn (M)} eventually strictly increases with strong exponential growth.

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[22] TATE J., Homology of Noetherian rings and local rings, Illinois J. Math. 1 (1957), 14–27.

AMS Subject Classification: 13D02, 13D40, 13H10, 13D25.

Eugene GOVER, Department of Mathematics, Northeastern University, Boston, MA 02115, U.S.A.



Syzygy 2005

R. Hartshorne

LIAISONWITH COHEN–MACAULAY MODULES


Abstract. We describe some recent work concerning Gorenstein liaison of codimension two

subschemes of a projective variety. Applications make use of the algebraic theory of maximal

Cohen–Macaulay modules, which we review in an Appendix.

1. Introduction

The purpose of this paper is to report on some recent work in the area of Gorenstein

liaison. For me this is a pleasant topic, because it illustrates the field of algebraic geom-

etry at its best. After all, algebraic geometry could be described as the use of algebraic

techniques in geometry and the use of geometric methods to understand algebra. In

the work I describe here, we found an unexpected connection between the theory of

maximal Cohen–Macaulay modules about which there is considerable algebraic liter-

ature, and the notion of Gorenstein liaison, which has emerged recently as geometers

attempted to generalize results about curves in P3 to varieties of higher codimension.

In Section 2, we review the “classical” case of curves in P3. In §3 we describegeneralizations of the notion of liaison to schemes of higher dimension and higher

codimension. Sections 4 and 5 develop the main new idea, which is instead of working

directly with schemes of codimension ≥ 3 in Pn , to consider subschemes of codimen-sion 2 of an arithmetically Gorenstein scheme X in Pn . Any liaison in X is also a

liaison in Pn , so this method is useful to establish existence of liaisons in Pn , but itcannot give negative results. We hope that the study of liaison on X may be interesting

in its own right, and give more insight into the nature of liaison in general.

Section 6 gives some applications, and Section 7 describes an interesting open

problem. The algebraic theory of maximal Cohen–Macaulay modules is reviewed in

an Appendix.

The principal new results described here are joint work with Marta Casanellas

and Elena Drozd, given in detail in the papers [3] and [4]. For background on liai-

son, I recommend the book of Migliore [16], and for information on Cohen–Macaulay

modules, the book of Yoshino [18].

It was a pleasure to attend the conference Syzygy 2005 in Torino in honor of

Paolo Valabrega’s sixtieth birthday, and I dedicate this paper respectively to him.

2. Curves in P3

We review the case of curves in P3, which has been known for some time, as a modelfor the more general situations that we will consider below.

419

420 R. Hartshorne

We work over an algebraically closed field k. A curve is a purely one-dimen-

sional scheme without embedded points. If C1 and C2 are curves in P3, we say they arelinked by a complete intersection curve Y , ifC1∪C2 = Y and ICi ,Y

∼= Hom(OC j,OY )

for i, j = 1, 2, i /= j . The equivalence relation generated by chains of linkages is

called liaison. If a liaison is accomplished by an even number of linkages, it is called

even liaison. To any curve C in P3 we associate its Rao module MC = H1∗ (IC ) =⊕n∈Z H1(P3, IC,P3(n)). The basic results about curves in P3 are the following

THEOREM 1 (Rao [17]). a) Two curves C1 and C2 are in the same even liaison

equivalence class if and only if their Rao modules M1,M2 are isomorphic, up to a shift

in degrees.

b) For any finite-length graded R = k[x0, x1, x2, x3]-module M, there exists a

nonsingular irreducible curve C in P3, whose Rao module is isomorphic to a shift ofM.

Thus the even liaison equivalence classes of curves in P3 are in one-to-onecorrespondence with finite length graded R-modules, up to shift.

For the next statement we need the notion of biliaison. If a curve C1 lies on a

surface S, and ifC2 ∼ C1+mH , meaning linear equivalence in the sense of generalizeddivisors [7] on S, where H is the hyperplane section of S, and m is an integer, then we

say that C2 is obtained by an elementary biliaison of height m from C1. If m ≥ 0 it is

an ascending elementary biliaison. It is easy to see that an elementary biliaison gives

an even liaison between C1 and C2. It is also easy to calculate numerical invariants

of C2, such as degree, genus, and postulation, from those of C1 in terms of m and the

degree of S.

THEOREM 2 (Lazarsfeld–Rao property [1], [15]). a) In any even liaison equiv-

alence class of curves in P3, the minimal curves (meaning those of minimal degree)form an irreducible family.

b) Any curve that is not minimal in its even liaison equivalence class can be

obtained by a sequence of ascending elementary biliaisons from some minimal curve.

REMARK 1. These results generalize well to subschemes V of codimension

two in Pn . The Rao module has to be replaced by a series of higher deficiency modulesHi∗(IV ) for 0 < i ≤ dim V and certain extensions between them: the best way to ex-

press this is by an element of the derived category. Or one can use the so-called E-type

resolution, in which case the set of even liaison equivalence classes of schemes V of

codimension two is in one-to-one correspondence with coherent sheaves E (satisfying

some additional conditions), up to stable equivalence and shift, and this in turn is in

one-to-one correspondence with the quasi-isomorphism classes of certain complexes

in the derived category replacing the Rao module.

The Lazarsfeld–Rao property also generalizes to codimension two subschemes

of quite general schemes. See for example [9] for precise statements and further refer-

ences.

Liaison with Cohen–Macaulay modules 421

3. Generalizations

When we consider curves in P4, or more generally, subschemes of codimension ≥ 3 in

any Pn , the direct analogue of Rao’s theorem fails. There are infinitely many distincteven liaison equivalence classes of curves all having the same Rao module, which can

be distinguished by other cohomological invariants [11]. It seems that liaison using

complete intersections, as we have defined it, is much too rigid to give an analogous

theory in higher codimension.

The notion of Gorenstein liaison seems to be a better candidate for generalizing

the theory.

DEFINITION 1. Two subschemes V1, V2 of Pn, equidimensional and withoutembedded components, are G-linked by an arithmetically Gorenstein scheme Y (mean-

ing the homogeneous coordinate ring of Y is a Gorenstein ring) if V1 ∪ V2 = Y and

IVi ,Y∼= Hom(OVj ,OY ) for i, j = 1, 2, i /= j . The equivalence relation generated

by chains of G-links is called Gorenstein liaison (or G-liaison for short), and if a G-

liaison can be accomplished by an even number of G-links, it is called even G-liaison.

It is easy to see for curves in Pn that even G-liaison preserves the Rao module(up to shift), as in the case of P3, and this naturally leads to the converse problem:

PROBLEM 3. If two curves in Pn have isomorphic Rao modules (up to shift),are they in the same even G-liaison class?

This problem is open at present. The special case when the Rao module is

zero is the case of arithmetically Cohen–Macaulay (ACM) curves, meaning that the

homogeneous coordinate ring is a Cohen–Macaulay ring. This includes in particular

the complete intersection curves. So the problem, which now can be stated for schemes

of any dimension is

PROBLEM 4. If V is an ACM scheme in Pn , is V in the Gorenstein liaison classof a complete intersection (glicci for short)?

This problem is also open at present, though many special cases are known (see

for example [11]). There are also candidates for counterexamples (as yet unproven),

such as 20 general points in P3, or a general curve of degree 20 and genus 26 in P4 [8].

Our approach in this paper, instead of studying the problem directly in Pn , willbe to study codimension two subscheme of an arithmetically Gorenstein variety X in

Pn . Liaisons in X can also be considered to be liaisons in Pn , and thus we study theproblem of higher codimension subschemes in Pn indirectly. While most of our resultsare valid for X of any dimension, for simplicity in this paper we will stick to dimension

3.

So here is the set-up. Let X be a fixed normal arithmetically Gorenstein sub-

variety of dimension 3 in Pn . We also keep fixed the embedding and hence the sheafOX (1) on X that defines the class of a hyperplane section H of X .

422 R. Hartshorne

If C1 and C2 are curves in X , we say that C1 and C2 are linked by a curve Y in X

if C1 ∪ C2 = Y and ICi ,Y∼= Hom(OC j

,OY ) for i, j = 1, 2, i /= j . If Y is a complete

intersection in X , meaning that Y is the intersection of surfaces defined by sections of

OX (a),OX (b) in X , then we say it is aC I -linkage. If Y is arithmetically Gorenstein (inthe ambient Pn), it is a G-linkage. These linkages give rise to the equivalence relationsof C I -liaison and even C I -liaison and G-liaison and even G-liaison as before.

Note that a C I -liaison in X is not necessarily a C I -liaison in Pn , unless X itselfis a complete intersection. However, a G-liaison in X is also a G-liaison in Pn .

If S is a surface in X containing a curve C , and if C ′ is another curve on S,with C ′ ∼ C + mH , meaning linear equivalence of generalized divisors on S, where

H is the hyperplane section, we say C ′ is obtained from C by an elementary biliaison

from C . If S is a complete intersection in X (corresponding to OX (a) for some a) itis a C I -biliaison. If S is an ACM scheme (in Pn) it is a G-biliaison. It is easy to seethat a C I -biliaison is an even C I -liaison. In fact, the equivalence relation generated

by C I -biliaisons is the same as even C I -liaison (proof similar to [7, 4.4]). One canshow also that a G-biliaison is an even G-liaison [11], [10, 3.6], however in generalthe equivalence relation generated by G-biliaisons is not the same as even G-liaison,

as we can see from the following example.

EXAMPLE 1. Let X be a nonsingular quadric hypersurface in P4. Every surfaceon X is a complete intersection, and in particular has even degree. Thus G-biliaisons

preserve the parity of the degree of a curve. On the other hand, the union of a rational

quartic curve with a line meeting it at two points is an arithmetically Gorenstein elliptic

quintic, so the two curves are G-linked. One line can also be linked to another line by

a conic, so we see that even G-liaison does not preserve parity of degree.

In studying G-liaison and G-biliaison on X , an important role is played by the

category of ACM sheaves on X . An ACM sheaf is a coherent sheaf E on X that is

locally Cohen–Macaulay and has vanishing intermediate cohomology: Hi∗(E) = 0 for

i = 1, 2. If X is P3, the only ACM sheaves are the dissocie sheaves, i.e., direct sums

of line bundles OX (ai ), by a theorem of Horrocks. However, if X is not P3, there areothers, and the category of these sheaves reflects interesting properties of X .

To see why these sheaves are important for G-liaison and G-biliaison, we first

mention the following result relating them to ACM surfaces in X and arithmetically

Gorenstein (AG) curves in X .

PROPOSITION 1. a) If S is an ACM surface in X, then its ideal sheaf IS,X is a

rank 1 ACM sheaf on X. Conversely if L is a rank 1 ACM sheaf on X, then for any

a ; 0, the sheaf L(−a) is isomorphic to the ideal sheaf IS,X of an ACM surface in

X.

b) If Y is an AG curve in X, then there is an exact sequence

0→ OX (−a) → N → IY,X → 0

for some a ∈ Z, where N is a rank 2 ACM sheaf on X with c1(N ) = −a. Converselyif N is any orientable (meaning c1(N ) = OX (a) for some a ∈ Z) rank 2 ACM sheaf


on X, and if s is a sufficiently general section of N (a) for a ; 0, then s induces an

exact sequence

0→ OXs→ N (a) → IY,X (b) → 0

for some AG curve Y in X and some b ∈ Z.

Proof. Part a) is elementary, while part b) is the usual Serre correspondence [4, 2.9].

Thus we see that the ACM surfaces and AG curves, which are used to define

G-biliaison and G-liaison, respectively, correspond in a natural way to rank 1 and rank

2 ACM sheaves on X . In the following two sections, we will study G-biliaison and

G-liaison separately.

4. Gorenstein biliaison

As in the previous section, we consider a normal arithmetically Gorenstein 3-fold X ,

and we will consider Gorenstein biliaison of curves on X .

First of all, let’s see what happens with a single elementary Gorenstein biliaison.

Let S be an ACM surface in X , let C be a curve in S, and let C ′ ∼ C+mH on S. Then

by construction, IC ′,S∼= IC,S(−m). Thus we can write exact sequences

0 → IS → IC ′ → IC ′,S → 0

‖0 → IS(−m) → IC (−m) → IC,S(−m) → 0.

If we let F be the fibered sum of IC ′ and IC (−m) over IC ′,S = IC,S(−m), we obtainsequences

0 → IS → F → IC (−m) → 0

0 → IS(−m) → F → IC ′ → 0.

Note here that the same coherent sheaf F appears in the middle of each sequence, and

that the sheaves on the left are rank 1 ACM sheaves on X that are isomorphic, up to

twist.

Conversely, given exact sequences

0 → L → F → IC (a) → 0

0 → L′ → F → IC ′(a′) → 0

with the same coherent sheaf F in the middle, where C,C ′ are curves in X, a, a′ inte-gers, and L,L′ rank 1 ACM sheaves that are isomorphic up to twist, it follows that C ′

is obtained by a single elementary G-biliaison from C . The idea of proof is to consider

the composed map L′ → F → IC (a). If this map is 0, then C ′ = C , which is a trivial

G-biliaison. If it is not zero, composing with the inclusion IC (a) ⊆ OX (a) identifiesL′(−a) with the ideal sheaf IS of an ACM surface on X and then one sees easily that

C ′ ∼ C + (a′ − a)H on S [3, 3.1, 3.3].

424 R. Hartshorne

With a little more work, one can arrive at an analogous criterion for two curves

to be related by a finite succession of elementary G-biliaisons.

THEOREM 3 ([3, 3.1]). Two curves C,C ′ on the normal arithmetically Goren-stein 3-fold X are in the same Gorenstein biliaison equivalence class if and only if

there exist exact sequences

0 → E → N → IC (a) → 0

0 → E ′ → N → IC ′(a′) → 0

with the same coherent sheaf N in the middle, where a, a′ are integers, and where Eand E ′ are ACM sheaves each having a filtration whose quotients are rank 1 ACM

sheaves (we call them layered ACM sheaves), and such that the rank 1 quotients of

these filtrations of E and E ′ are isomorphic up to order and twists.

The point is that each G-biliaison contributes a rank 1 ACM factor, but that

these make up the two sheaves E and E ′ in a different order, and with different twistsassociated to each.

If E is a layered ACM sheaf as above, the filtration with rank 1 ACM quotients

may not be unique. Taking advantage of this are two “exchange lemmas” [3, 3.4, 4.6]that allow one to replace one E by another E ′ having the same factors, in sequences asin Theorem 3, after passing to another curve in the same G-biliaison class. These form

a sort of converse to Theorem 3, and allow us to formulate a necessary and sufficient

condition for the property analogous to Problem 4 on X , namely that every ACM curve

on X should be in the G-biliaison class of a complete intersection on X . This condition

is a bit complicated to state (see [3, 4.2, 4.3]), so instead here we will explain the resultonly in one interesting special case.

THEOREM 4 ([3, 6.2]). Let X be the cone over a nonsingular quadric surface

in P3. (Thus X is a normal quadric hypersurface in P4 having one double point.) Thentwo curves C and C ′ on X are in the same Gorenstein biliaison equivalence class if and

only if their Rao modules are isomorphic, up to shift. In particular, all ACM curves

are equivalent for G-biliaison.

Idea of Proof. It is obvious that Gorenstein biliaison preserves the Rao module, up to

shift, so one direction is clear.

For the other direction, let C be any curve in X , with Rao module M . Our

strategy is to construct another curve C ′ that depends only on M , and then show that Cand C ′ are in the same G-biliaison equivalence class, which will prove the theorem.

Given M , let M∗ be the dual module, and take a resolution

0→ G → F2 → F1 → F0 → M∗ → 0

over R, the homogeneous coordinate ring of X , where the Fi are free graded R-

modules, and G is the kernel. Let N ′ be the sheaf associated to G∨, and let L′ be


a dissocie sheaf of rank one less mapping toN ′ so as to define a curve C ′ by its coker-nel:

0→ L′ → N ′ → IC ′(a′) → 0.

On the other hand, let

0→ L → N → IC → 0

be an N -type resolution of C , i.e., with L dissocie and N coherent, locally Cohen–

Macaulay, and H1∗ (N ) ∼= M and H2∗ (N ) = 0. Let N = H0∗ (N ) and take a resolution

0→ P → L1 → L0 → N → 0

over R with Li free and P the kernel. Dualizing gives an exact sequence

0→ N∨ → L∨0 → L∨

1 → P∨ → M∗ → 0.

Now there is a natural map of the earlier free resolution of M∗ into this one, andthis gives us a map of G to N∨, from which we obtain a natural map N → N ′. Byadding extra free factors if necessary, we may assume it is surjective, and then let E be

the kernel:

0→ E → N → N ′ → 0.

Since N and N ′ both have H1∗ = M and H2∗ = 0, we see that E is an ACM sheaf

on X . Furthermore, taking the composed map from N to IC ′(a′) we obtain an exactsequence

0→ E ⊕ L → N → IC ′(a′) → 0.

In order to apply the criterion of Theorem 3 we now need to use the special

property of the quadric 3-fold X (see Appendix), which tells us first that every ACM

sheaf on X is layered, secondly that the only rank 1 ACM sheaves on X (up to twist)

are OX , ID , and IE , where D, E represent the two types of planes in X , and thirdlythat there is an exact sequence

0→ ID → O2X → IE (1) → 0.

In the ACM sheaf E , copies of ID and IE (and their twists) must occur in equal

numbers, because E is orientable. Then the exchange lemmas referred to above allow

us to replace an ID plus an IE by an O2X . Thus E ⊕ L is replaced by a dissocie sheaf,

and then Theorem 3 tells us that C and C ′ are in the same G-biliaison class. (For moredetails see [3, 4.7, 6.2].)

5. Gorenstein liaison

Let us consider a normal AG 3-fold X , as before, and study Gorenstein liaison equiva-

lence of curves in X . Since the AG curves in X are associated to rank 2 ACM sheaves

on X , as we saw above, we expect to see them play a role.

First of all, let us see what happens with a single Gorenstein liaison. We track

this behavior using the N -type resolution of a curve C .

426 R. Hartshorne

PROPOSITION 2. Let C be a curve in X with N -type resolution 0 → L →N → IC → 0, and suppose that C is linked to a curve C ′ by the AG curve Y . Then

C ′ has an N -type resolution of the form

0→ L′ → N ′ → IC ′(a′) → 0

with L′ dissocie, and where N ′ is an extension

0→ L∨ ⊕ E∨ → N ′ → N σ∨ → 0,

where E is the rank 2 ACM sheaf associated to Y , and N σ∨ denotes the dual of the

first syzygy sheaf of N .

To prove this (see [4, 3.2]) one first uses the usual cone construction of the mapIY ⊆ IC , and this gives the sequence

0→ N∨ → L∨ ⊕ E∨ → IC ′(a) → 0.

This is not an N -type resolution, but by using the syzygy sheaf N σ of N , one can

transform it into the desired N -type resolution.

Note what happens to the Rao module M . From the definition of the syzygy

sheaf

0→ N σ → F → N → 0

with F dissocie, we see that M ∼= H1∗ (N ) ∼= H2∗ (N σ ). By Serre duality then

H1∗ (N σ∨) ∼= M∗, the dual of M , and this shows that the Rao module of C ′ is M∗

shifted, as we would expect from a single liaison.

This proposition shows us that a single G-liaison complexifies the N -type res-

olution by throwing in a dual of a syzygy, and adding an extension by a rank 2 ACM

sheaf. There is a sort of converse to this, showing how to simplify anN -type resolution

by removing a rank 2 ACM sheaf. In general, this cannot be accomplished by a single

G-liaison, but requires a more complicated procedure.

PROPOSITION 3. Let C be a curve with an N -type resolution

0→ L → N → IC → 0,

and suppose given an exact sequence

0→ E → N → N ′ → 0

with E a rank 2 ACM sheaf and N ′ a locally CM sheaf of rank ≥ 2. Then there

is a curve C ′ in the same even G-liaison equivalence class as C having an N -type

resolution

0→ L′ → N ′ → IC ′(a′) → 0.

Proof. See [4, 3.4].


Using these two propositions, it is possible to give a criterion, in terms of the

N -type resolutions, for when two curves are in the same G-liaison class [4, 5.1]. Theexact statement, which involves successive extensions by rank 2 ACM sheaves and

their syzygy duals (which may no longer be of rank 2), is rather complicated, so we

omit it here. Using this theorem, one can also give a criterion for every ACM curve to

be in the Gorenstein liaison class of a complete intersection [4, 5.4]. Here we will justgive one special case, albeit an interesting one.

THEOREM 5. Let X be a nonsingular quadric hypersurface in P4. Then twocurves are in the same even G-liaison class if and only if their Rao modules are iso-

morphic, up to shift.

Sketch of Proof (cf. [4, 6.2]). Let C be any curve, with Rao module M , and let C ′ beanother curve with the same Rao module M , constructed as in the proof of Theorem 4

above. Following the plan of that proof we have N -type resolutions

0→ L → N → IC → 0

0→ L′ → N ′ → IC ′(a′) → 0

and an exact sequence

0→ E → N → N ′ → 0

where E is an ACM sheaf on X .

Now we invoke the special property of the nonsingular quadric 3-fold X , which

is that every ACM sheaf is a direct sum of a dissocie sheaf and copies of twists of

a single rank 2 ACM sheaf E0, associated to a line in X (see Appendix). We apply

Proposition 3 repeatedly to remove copies of E0 and its twists from N , thus eventually

obtaining a curve C ′′, in the same even G-liaison class as C , and having an N -type

resolution whose middle sheaf N ′′ differs from N ′ only by a dissocie sheaf. Then N ′

and N ′′ are stably equivalent, and so C ′′ and C ′ are in the same even C I -liaison class,by Rao’s theorem, and a fortiori in the same even G-liaison class.

Note that in the case of the nonsingular quadric 3-fold, G-biliaison is just the

same as C I -biliaison, hence is much too restrictive to provide a result like this theorem.

6. Applications

In [11, 8.10] the authors showed, by an exhaustive listing of all possible ACM curves

on these surfaces, that any ACM curve lying on a general smooth rational ACM surface

in P4 is glicci. The rational ACM surfaces in P4 (not counting those in P3, for whichthe theorem is known) are the cubic scroll, the Del Pezzo surface of degree 4, the

Castelnuovo surface of degree 5, and the Bordiga surface of degree 6.

For the cubic scroll, the Del Pezzo, and the Castelnuovo surface, this result is an

immediate consequence of our Theorem 4, because each of these surfaces is contained

in a quadric 3-fold with one double point. Our method does not apply to the Bordiga

surface, which is not contained in any quadric hypersurface.

428 R. Hartshorne

In his paper [14], Lesperance studied curves in P4 of the following form. Let Cbe the disjoint union C1∪C2 of two plane curves C1,C2, lying in two planes that meetat a single point P . Let them have degrees d1, d2, and assume either a) 2 ≤ d1 ≤ d2 or

b) 2 ≤ d1 and C2 contains the point P . The Rao module is then M ∼= R/(IP + R≥d1),which depends only on the point P and the integer d1. Lesperance shows that all the

curves of type a) and some of those of type b) are in the same Gorenstein liaison class,

by using explicitly constructed G-liaisons.

Since a union of two planes meeting at a point is contained in a quadric hyper-

surface with one double point, it follows from our Theorem 4 that all the above curves

with the same Rao module are equivalent for G-liaison [3, 6.4].

A third application is the following

THEOREM 6. Any arithmetically Gorenstein scheme V in Pn is in the Goren-stein liaison class of a complete intersection (glicci).

For the proof [4, 7.1] we use the higher-dimensional analogues of the resultsdescribed in this paper for an AG 3-fold X . By a Bertini-type theorem of Altman

and Kleiman, one can find a complete intersection scheme X in Pn , containing V , ofdimension two greater than V , and smooth outside of V . Then X is normal and AG,

and V is a codimension two AG scheme in X , so there is an exact sequence

0→ OX (−a) → E → IV,X → 0

where E is a rank 2 ACM sheaf on X . Let M be a rank 2 dissocie sheaf on X and

consider the new N -type resolution of V ,

0→ OX (−a) ⊕M → E ⊕M → IV,X → 0.

Then we apply the analogue of Proposition 3, which is [4, 3.4], to remove Eand obtain another subscheme V ′ ⊆ X , in the same even G-liaison class as V , with an

N -type resolution

0→ L′ → M → IV ′,X (a′) → 0.

SinceM is rank 2 dissocie, it follows that V ′ is a complete intersection in X , and sinceX is itself a complete intersection in Pn , V ′ is also a complete intersection in Pn . SinceG-liaisons in X are also G-liaisons in Pn , we find that V is glicci, as required.

7. An open problem

If there is a moral to all the investigations of Gorenstein liaison so far, it seems to

me that good results are obtained for schemes with some special structure, such as

determinantal schemes [11], or schemes of codimension 2 in low-degree hypersurfaces,

such as the ones considered in Sections 4,5 above.

To describe a situation on the border between what is known and what is not

known, I would like to consider the case of zero-dimensional subschemes of a non-

singular cubic surface in P3. Though of one dimension lower than the discussions


earlier in this paper, I think it is a good arena to test the essential difficulties of the

subject.

So, let X be a nonsingular cubic surface in P3. We consider zero-schemesZ ⊆ X . Any zero-scheme is ACM, so there are two problems to consider.

PROBLEM 5. Is every zero-scheme Z ⊆ X in the G-biliaison equivalence class

of a point?

PROBLEM 6. Is every zero-scheme Z ⊆ X in the G-liaison equivalence class

of a point?

Both problems are open at present. I will discuss what is known about them so

far.

Using explicit G-liaisons and G-biliaisons on ACM curves on X , one can show

that any set Z of n points in general position on X is G-liaison equivalent to a point [8,

2.4]. The proof of this result is curious, in that one uses sequences of liaisons wherethe number of points may have to increase before it decreases. For example, starting

with 18 general points, one makes links to the following numbers of points (always in

general position): 18 → 20 → 28 → 22 → 16 → 13 → 7 → 5 → 3 → 1. For

points in special position, it seems hopeless to generalize this method.

Another approach, more in the spirit of this paper, is to study the category of

ACM sheaves on X . Faenzi [6] has classified the rank 2 ACM sheaves on X . Up to

twist, there is a finite number of possible Chern classes, and for fixed Chern classes,

the possible sheaves form algebraic families of dimensions ≤ 5. Already the presence

of families of dimension > 1 shows that we are in a situation of “wild CM-type” (see

Appendix). Looking at Faenzi’s results, again it seems hopeless to achieve a complete

classification of ACM sheaves of all ranks on X . One can show, however, that there

are families of arbitrarily high dimension of indecomposable ACM sheaves of higher

rank.

However, to answer the two problems above, one would not need a complete

classification of ACM sheaves on X . For an affirmative answer to Problem 5, it would

be sufficient to show [3, 4.3].

(∗) Every orientable ACM sheaf E on X has a resolution

0→ F2 → F1 → E → 0

where F1 and F2 are layered ACM sheaves (i.e., successive extensions of rank 1 ACM

sheaves).

In regard to this property, there are examples of rank 2 ACM sheaves E on X ,

that are not layered themselves, but do have a resolution of this form. So there seems

to be some hope that this may hold.

For an affirmative answer to Problem 6, it would be sufficient to show [4, 5.4].

(∗∗) Every orientable ACM sheaf on E is stably equivalent to a double-layered

sheaf on X (which is a successive extension of rank 2 ACM sheaves and their syzygies).

430 R. Hartshorne

Appendix. MCM modules and ACM sheaves

In this appendix we give a brief outline of some algebraic results that are needed to jus-

tify the results on ACM sheaves on quadric hypersurfaces used in Sections 4,5 above.

Let R,m be a Cohen–Macaulay local ring. A maximal Cohen–Macaulay

(MCM) module is a finitely generated R-module M , with depth M = dim R and

Supp M = Spec R.

For example, if R is a regular local ring, every MCM module has homological

dimension zero, and so is free. Conversely, if R,m is a Cohen–Macaulay local ring

over which every MCM is free, then R is regular. Indeed, for any R-module N , con-

sider a free resolution of length n = dim R, and let M be the kernel at the last step.

Then M is an MCMmodule, hence free, and so N has a finite free resolution. Thus the

ring R has finite global homological dimension, and by a theorem of Serre, this implies

that R is regular.

Thus the presence of non-trivial MCM modules characterizes non-regular local

rings, and the category of MCM modules is an interesting measure of the complexity

of the singularity of the local ring.

In certain circumstances, Y. Drozd [5] has shown that local rings can be divided

into three classes, depending on the behavior of the MCM modules. It is true in any

case that an MCM module can be written uniquely as a direct sum of indecomposable

MCM modules, namely those that allow no further direct sum decomposition. We say

that R is of finite CM-type if there is only a finite number of indecomposable MCM

modules. We say R is of tame CM-type if the indecomposable MCM modules form a

countable number of families of dimension at most one. We say R is of wild CM-type

if there are families of arbitrarily large dimension of indecomposable MCM modules.

The tame-wild dichotomy theorem says (in certain cases) that only these cases can

occur. While to my knowledge this has not been proved in general, we can keep it in

mind as a principle of what to expect when studying MCM modules.

The same definitions apply to the case of graded rings and graded modules, and

thus admit a translation into sheaves on projective schemes. If X is a ACM scheme

in Pn , we have defined an ACM sheaf on X to be a locally Cohen–Macaulay coherent

sheaf E on X with no intermediate cohomology: Hi∗(E) = 0 for 0 < i < dim X . If

R is the homogeneous coordinate ring of X , then we obtain a correspondence between

ACM sheaves on X and gradedMCMmodules on R by sending a sheaf E to the module

E = H0∗ (E), and sending the module E to the associated sheaf E = E . In carrying

over results and definitions from the local case, we should consider graded modules up

to shift, and ACM sheaves up to twist. So we can say X is of finite CM-type if there

is only a finite number (up to twist) of isomorphism classes of indecomposable ACM

sheaves.

To illustrate the different CM-types in the projective case, note that if X is a

nonsingular curve in Pn , then an ACM sheaf on X is just a locally free sheaf E , also

called a vector bundle. If X is rational, of degree d, the only indecomposable vector

bundles (up to twist by OX (1)) are line bundles of degrees 0 ≤ e < d, so X is of finite

CM-type. If X is an elliptic curve, then by the classification theorem of Atiyah, for each


rank and degree (mod d = deg X) there is a one-parameter family of isomorphismclasses of indecomposable vector bundles of rank r and degree e. Thus X is of tame

CM-type. And if the genus of X is g ≥ 2, then as the rank grows, so does the dimension

of the moduli space of stable vector bundles, so X is of wild CM-type.

In the complex-analytic and complete local ring case, those local rings of iso-

lated hypersurface singularities of finite CM type have been classified [13], [2]. They

are the local rings of simple singularities in the sense of Arnol’d; in each dimension

they are associated with Dynkin diagrams An, Dn, E6, E7, E8, and their equations canbe written explicitly.

Carrying these results over to the graded case, one obtains a list of all projective

schemes of finite CM-type [18], namely, projective spaces, nonsingular quadric hyper-

surfaces in any dimension, the rational cubic scroll in P4, and the Veronese surface inP5. Furthermore, the indecomposable ACM sheaves on these varieties can be described

explicitly, and this is where we find that there is just one non-trivial indecomposable

ACM sheaf on the nonsingular quadric 3-fold, mentioned in the proof of Theorem 5.

The main tools for studying MCM modules on hypersurface singularities, or

ACM sheaves on hypersurfaces in Pn , are the matrix factorization, and the doublebranched covers and periodicity theorems of Knorrer [13]. We explain these in the

projective case.

Let X be a hypersurface in Pn , and let E be an ACM sheaf on X . Since the

associated graded module E = H0∗ (E) has depth n over the coordinate ring P =k[x0, . . . , xn] of Pn , there is a resolution

0→ L1ϕ→ L0 → E → 0

by dissocie sheaves Li on Pn of the same rank m. This gives a square matrix ϕ ofhomogeneous forms in P . Then one shows that there is another matrix ψ of the same

rank, with the property that ψ · ϕ = ϕ · ψ = f · id, where f is the equation of the

hypersurface. This is called a matrix factorization of f . One sees also that det ϕ = f r ,

where r = rank E . These constraints allow one to gain information about the possible

ACM sheaves E when the numbers are small enough.

The other technique is Knorrer’s double branched cover, and periodicity theo-

rems, which allow one to pass from a hypersurface X in Pn defined by a polynomialf ∈ P to the hypersurface X ′ in Pn+1 defined by f + x2, or the hypersurface X ′′ inPn+2 defined by f + x2 + y2, where x and y are new variables.

In the paper [3], we use these techniques to show that the singular quadric 3-fold

X in P4 with one double point is of countable CM-type, namely it has only countablymany indecomposable ACM sheaves (up to twist), and these are OX , ID, IE , whereD, E are the two types of planes in X , and two infinite sequences E" and E

′", for " =

1, 2, . . . , of rank 2 ACM sheaves that are each extensions of suitable twists of ID and

IE [3, 6.2], hence layered. This is the result needed for the proof of Theorem 4 above.

A good reference for the material described in this appendix, besides the original

papers, is the survey article [12] and the book of Yoshino [18].

432 R. Hartshorne

References

[1] BALLICO E., BOLONDI G. AND MIGLIORE J.C., The Lazarsfeld–Rao problem for liaison classes of

two-codimensional subschemes of Pn , Amer. J. Math. 113 (1991), 117–128.

[2] BUCHWEITZ R.-O., GREUEL G.-M. AND SCHREYER F.-O., Cohen–Macaulay modules on hyper-

surface singularities. II, Invent. Math. 88 (1987), 165–182.

[3] CASANELLAS M. AND HARTSHORNE R., Gorenstein biliaison and ACM sheaves, J. Algebra 278

(2004), 314–341.

[4] CASANELLAS M., DROZD E. AND HARTSHORNE R., Gorenstein liaison and ACM sheaves, J. Reine

Angew. Math. 584 (2005), 149–171.

[5] DROZD Y.A., Cohen–Macaulay modules over Cohen–Macaulay algebras, Canadian Math. Soc. Conf.

Proc. 19 (1996), 25–52.

[6] FAENZI D., Rank 2 arithmetically Cohen–Macaulay bundles on a nonsingular cubic surface,

math.AG/0504492, preprint.

[7] HARTSHORNE R., Generalized divisors on Gorenstein schemes, K -Theory 8 (1994), 287–339.

[8] HARTSHORNE R., Some examples of Gorenstein liaison in codimension three, Collect. Math. 53

(2002), 21–48.

[9] HARTSHORNE R., On Rao’s theorem and the Lazarsfeld–Rao property, Ann. Fac. Sci. Toulouse 12 (3)

(2003), 375–393.

[10] HARTSHORNE R., Generalized divisors and biliaison, math.AG/0301162, preprint.

[11] KLEPPE J., MIGLIORE J.C., MIRO–ROIG R.M., NAGEL U. AND PETERSON C., Gorenstein Liaison,

Complete Intersection Liaison Invariants, and Unobstructedness, Mem. Amer. Math. Soc. 154 (2001),

no. 732.

[12] KNORRER H., Cohen–Macaulay modules on hypersurface singularities, in: “Representations of Alge-

bras (Durham, 1985)”, London Math. Soc. Lecture Notes Ser. 116, Cambridge University Press 1986,

147–164.

[13] KNORRER H., Cohen–Macaulay modules on hypersurface singularities. I, Invent. Math. 88 (1987),

153–164.

[14] LESPERANCE J., Gorenstein liaison of some curves in P4, Collect. Math. 52 (2001), 219–230.

[15] MARTIN–DESCHAMPS M. AND PERRIN D., Sur la classification des courbes gauches, Asterisque

184–185, 1990.

[16] MIGLIORE J.C., Introduction to liaison theory and deficiency modules, Birkhauser, Boston 1998.

[17] RAO A.P., Liaison among curves in P3, Invent. Math. 50 (1979), 205–217.

[18] YOSHINO Y., Cohen–Macaulay modules over Cohen–Macaulay rings, London Math. Soc. Lecture

Notes Ser. 146, Cambridge University Press 1990.

AMS Subject Classification: 14M06, 14M07, 13C40, 13C14.

Robin HARTSHORNE, Department of Mathematics, University of California, Berkeley, CA 94720-3840,

USA



Syzygy 2005

S. Nollet

DEFORMATIONS OF SPACE CURVES: CONNECTEDNESS OF

HILBERT SCHEMES


Abstract. We survey the Hilbert schemes Hd,g of Cohen-Macaulay space curves having

degree d and genus g, giving their geography and the current state of the connectedness

problem. Focusing on a specific example, we then describe the irreducible families of curves

in H4,−99 and explain the connectedness, paying special attention to certain deformationson the double quadric surface. We close with some new results, determining which families

of degree four curves are subcanonical and showing how some examples of Chiantini and

Valabrega fit into this classification.

1. Introduction

Early in the development of scheme theory in algebraic geometry, Grothendieck con-

structed the fine moduli space for flat families of subschemes in Pn , known as theHilbert scheme [15]. Since the Hilbert polynomial is constant for flat families over a

connected base, the Hilbert scheme Hilbn can be written as a disjoint union of pieces

Hilbnp(z) indexed by the corresponding Hilbert polynomials. As a fine moduli space,

these schemes come equipped with universal flat family

(1)

X ⊂ Hilbnp(z) × Pn

↓Hilbnp(z)

having fibres with Hilbert polynomial p(z) such that for any flat family

(2)

Y ⊂ T × Pn

↓T

with fibres of Hilbert polynomial p(z), there is a unique map T → Hilbnp(z) such that

diagram (2) is obtained from diagram (1) by pull-back. Thus one studies the Hilbert

scheme by producing flat families. As Grothendieck showed that Hilbnp(z) is projective

over SpecZ, the set of all projective subschemes is encoded by equations with integercoefficients.

Since flat families over a connected base have constant Hilbert polynomial, it’s

natural to ask whether the converse is true: given two subschemes in Pn with thesame Hilbert polynomial, is there a connected flat family of which both are a mem-

ber? Equivalently, is the Hilbert scheme connected? This was answered by Hartshorne

in his PhD thesis [19].

433

434 S. Nollet

THEOREM 1 (Hartshorne, 1962). For any p(z) ∈ Q[z] and any field k, theHilbert scheme Hilbnp(z) for closed subschemes X ⊂ Pnk with Hilbert polynomial p(z)is connected whenever it is non-empty.

The geography is an important aspect of any moduli problem: for which natural

invariants of the problem is the moduli space non-empty? There are at least three

characterizations of the polynomials p(z) ∈ Q[z] for which there is a subscheme V ⊂Pn having Hilbert polynomial p(z). One follows from Macaulay’s theorem on the

growth of the Hilbert function of a standard k-algebra [17], another is a consequence

of Hartshorne’s thesis [19] and a third occurs naturally from Green’s interpretation of

Macaulay’s bound in terms of restricted linear series [14]: a summary and comparison

is given in [4].

We now specialize to space curves: take n = 3 and let Hilbd,g denote the Hilbert

scheme of subschemes in P3 with Hilbert polynomial p(z) = dz+ 1− g, the curves of

degree d and arithmetic genus g. Classically one is interested in the open subscheme

H0d,g ⊂ Hilbd,g

corresponding to smooth connected curves. The geography for this problem (the pairs

(d, g) for which H0d,g is non-empty) was known to Halphen and completely proved

by Gruson and Peskine a hundred years later [16]. As to connectedness, we have the

following results of Harris [18] and Ein [10].

THEOREM 2 (Harris, 1982). H0d,g is irreducible if d ≥ 54g + 1.

THEOREM 3 (Ein, 1986). H0d,g is irreducible if d ≥ g + 3.

EXAMPLE 1. The Hilbert schemes H0d,g are not connected in general: the

smallest example is H09,10 [20, IV, Ex. 6.4.3], which has two connected components,

the curves of type (3, 6) on a smooth quadric and complete intersections of two cubics.More generally, H0d,g is not connected for d ≥ 9 and g = 2d − 8. Indeed, the curvesC of type (3, d − 3) on a smooth quadric satisfy h0OC (2) = 9 and h0IC (2) = 1 while

curves D not lying on a quadric satisfy h0OD(2) ≥ 10 and h0ID(2) = 0. By semi-

continuity, it follows that the curves of type (3, d − 3) form a connected component

of H0d,2d−8. Note that there exist other components, as such curves exist on a cubic

or quartic surface. Guffroy conjectures that H0d,g is irreducible for g < 2d − 8 (i.e.

d > 12g + 4) and proves it for d ≤ 11 [17]. If true, the conjecture would strongly

improve the results above.

The subject of this survey is yet a third moduli space, namely the Hilbert scheme

of locally Cohen-Macaulay curves without isolated points, the pure one-dimensional

subschemes of P3 of degree d and genus g. Following Martin-Deschamps and Perrin[27, 29], we denote these Hilbert schemes by Hd,g , which sit between the two extremes

considered above:

H0d,g ⊂ Hd,g ⊂ Hilbd,g.

Deformations of space curves 435

The Hilbert schemes Hd,g are natural from the perspective of liaison theory, which has

seen a great deal of activity over the last 25 years: Migliore’s book [31] provides an

excellent survey of this work. The point is that liaison preserves the property of being

locally Cohen-Macaulay [31, Cor. 5.2.12] but does not preserve geometric properties

such as smoothness, irreducibility, or reducedness. On the other hand, even the most

general locally Cohen-Macaulay curves can be brought to the classical curves through

a sequence of liaisons, as proved by Rao [38, Thm. 2.6].

THEOREM 4 (Rao, 1979). Every liaison class contains a smooth connected

curve.

Thus the schemes Hd,g are the result of starting with the smooth connected

curves and closing off under the equivalence relation of liaison. In view of the connec-

tivity results above, the following question is natural:

PROBLEM 7. For which pairs (d, g) is Hd,g connected?

REMARK 1. This does not follow in any easy way from the proof of Theorem

1, as Hartshorne constructs deformations which typically pass through (non-reduced)

subschemes having embedded points. The real question here is whether curves with

embedded points can be avoided.

In addressing the status of Problem 7, we begin with the geography of locally

Cohen-Macaulay space curves in §2. This includes (a) the determination of the pairs

(d, g) for which Hd,g is non-empty and (b) the cohomological bounds leading to the

special families of extremal and subextremal curves. The extremal curves become

prominent in §3 when we give connectedness results for the Hilbert schemes. We

follow this up with an example in §4, describing all the irreducible components of

the Hilbert scheme H4,−99 and explaining why this scheme is connected. In §5 wediscuss deformations of curves on a double surface and show how a disjoint union of

two double lines can be deformed to a multiplicity four line without adding embedded

points, a crucial part of the proof that H4,−99 is connected. Finally, in §6 we determinewhich families of degree four curves are sub-canonical. In particular, we show how

examples of Chiantini and Valabrega [6, Ex. 3.1 and 3.2] fit into our classification.

The author thanks E. Cabral Balreira for his help with making the figures and

Mario Valenzano for corrections on the first draft.

2. The geography of Cohen-Macaulay curves

In this section we describe the pairs (d, g) for which our Hilbert schemes Hd,g are non-

empty. As a byproduct of the proof, we will encounter the extremal curves, which play

an important role in the following section. The starting point is the following theorem

[28, Thm. 2.5 and Cor. 2.6].

THEOREM 5 (Martin-Deschamps and Perrin, 1993). Assume char k = 0. If

436 S. Nollet

C ∈ Hd,g is non-planar, then the Rao function h1IC (n) is bounded by the function

depicted in Figure 1. In particular, g ≤(d−22

).

nd 2

d 2

2g( )

Figure 1: Bound of Theorem 5 on h1IC (n) for non-planar curves

This generalizes to curves in higher dimensional projective space [8], though

the bounding function is more complicated. The characteristic zero hypotheses is used

to prove that if C is a curve of degree d ≥ 3 not contained in a plane, then the general

hyperplane section H ∩ C is not contained in a line. While this fails in characteristic

p > 0 [21, Ex. 2.3], the bound on cohomology still holds [32, Prop. 2.1], as does the

bound on the genus [21, Cor. 3.6]:

THEOREM 6 (Hartshorne 1994). The Hilbert scheme Hd,g is non-empty if and

only if either

(a) d ≥ 1 and g =(d−12

), or

(b) d ≥ 2 and g ≤(d−22

).

One way to prove that Hd,g is non-empty for g ≤(d−22

)is to observe that there

are curves which achieve equality in Theorem 5 [29, Prop. 0.5]:

THEOREM 7 (Martin-Deschamps and Perrin, 1996). For all d ≥ 2 and g ≤(d−22

)there are curves C ∈ Hd,g giving equality in Theorem 5 for all n.

The curves of Theorem 7 are called extremal curves and have some interesting

properties. For example, the subset of extremal curves forms an irreducible component

E ⊂ Hd,g [29, Thm. 3.7], which is non-reduced except when d = 2 (double lines),

g =(d−22

)(ACM extremal curves) or d = 3 and g = −1 [29, Thm. 5.3].

REMARK 2. The following are equivalent:


1. C is an extremal curve.

2. C is a minimal curve for a complete intersection module annihilated by two

linear forms (this allows one to write the total ideal and minimal resolutions for

extremal curves [29, Prop. 0.5, 0.6 and Thm. 1.1]).

3. C is non-planar of degree d and contains a planar subcurve of degree d − 1 ([11,§2, Thm. 8] or [32, Prop. 2.2]).

Assuming char k = 0, Ellia observed [11, §2, Prop. 9] that a curve which is

neither planar nor extremal satisfies even stronger bounds on the Rao function. Using

Schlesinger’s spectrum of a curve [40], this bound was refined while removing the

characteristic zero hypothesis [32, Thm. 2.11]:

THEOREM 8 (Ellia and Nollet, 1997). If C ∈ Hd,g is a non-planar and non-

extremal, then the Rao function h1IC (n) is bounded by the function depicted in Figure2. In particular, g ≤

(d−32

)+ 1.

n

d 3

2g+1

d 31

Figure 2: The bound of Theorem 8 on h1IC (n) for non-extremal curves

A curve C ∈ Hd,g is subextremal if it achieves the bound of Theorem 8 for

all n. A curve C ∈ Hd.g is subextremal if and only if it is a height one elementary

biliaison of an extremal curve C ′ ∈ Hd−2,g+3−d on a quadric surface [32, Thm. 2.14]

and hence exist for all d ≥ 4 and g ≤(d−32

)+ 1: letting S ⊂ Hd,g denote the family

of subextremal curves, the universal biliaison scheme of Martin-Deschamps and Perrin

shows that S is irreducible. Indeed, if E ⊂ Hd−2,g+3−d is the extremal component, wecan consider the set B of triples (C,C ′, Q) for which C is a height one biliaison of C ′

on the quadric surface Q. The natural projections

Bp1→ S

p2↓E

438 S. Nollet

are smooth and irreducible [27, VII, §4], hence irreducibility of E implies irreducibility

of S.

REMARK 3. Given Theorem 5 and Theorem 8, one might expect that curves

which are neither planar nor extremal nor subextremal should satisfy even stronger

bounds. This fails, however: there are curves which give equality in Theorem 8 for

some values of n, but not others [32, Ex. 2.15 and 2.17].

REMARK 4. As the extremal curves form an irreducible component, one might

expect that the closure of the subextremal curves S ⊂ Hd,g to form an irreducible

component as well (though S itself is not closed: its closure contains extremal curves

[34]). Uwe Nagel has informed me that this is indeed true and is current joint work

between he, Nadia Chiarli and Silvio Greco.

d

g

2 3 41

d 1

2g =

d 2

2g = d 3

2g = +1

Figure 3: The geography for locally Cohen-Macaulay curves

3. Connectedness results

In this section we summarize the current state of Problem 7. We will begin with some

general results about families of curves that can be deformed to extremal curves and

then proceed to particular ranges. In terms of the geography of Cohen-Macaulay curves

(Figure 3), we will see in Theorems 10 and 11 that Hd,g is connected for pairs (d, g)near the boundaries at the top and to the left.

Many families of curves can be deformed to extremal curves (without passing

through curves with embedded points).


THEOREM 9. The following families of curves can be deformed in Hd,g to ex-

tremal curves.

(1) Disjoint unions of lines.

(2) Smooth rational curves.

(3) Smooth connected curves with d ≥ g + 3.

(4) ACM curves.

(5) The disjoint union of an extremal curve and a line.

(6) The union of an extremal curve and a line meeting at a point.

(7) Any curve in the liaison class of an extremal curve.

Proof. (1)-(6) are results of Hartshorne [22] and (7) is due to Perrin [18].

When the arithmetic genus g is large relative to the degree d, the Hilbert scheme

Hd,g has few irreducible components, making it relatively easy to check connectedness.

The following result is the work of several authors.

THEOREM 10. If g ≥(d−32

)− 1, then Hd,g is connected.

Proof. According to Theorem 6, either g =(d−12

)(in which case Hd,g is the irreducible

family of plane curves) or g ≤(d−22

). In the range

(d−32

)+ 1 < g ≤

(d−22

), Theorem

8 shows that Hd,g = E is the family of extremal curves, which is irreducible by the

work of Martin-Deschamps and Perrin [29].

There are three more arithmetic genera to check, but things become more deli-

cate, as Hd,g is not irreducible.

If g =(d−32

)+ 1, then Theorem 8 shows that each curve C ∈ Hd,g is ex-

tremal or ACM, since the bound on h1IC (n) is zero. Conversely each ACM curve in

Hd,g is subextremal by definition, hence Hd,g = E ∪ S consists only of extremal and

subextremal curves. Finally E ∩ S /= ∅ by [34] and Hd,g is connected.

If g =(d−32

), then the non-extremal curves C satisfy h1IC (n) ≤ 1. Samir

Aıt-Amrane showed [1] that Hd,g has three irreducible components for large d: (a)

extremal curves, (b) subextremal curves and (c) bilinks of height one from a double

line of genus −1 on a surface of degree d − 2. Both families (b) and (c) specialize

to family (a) by Theorem 9(7), but Samir’s method was to use the triads developed by

Hartshorne, Martin-Deschamps and Perrin [23].

If g =(d−32

)− 1, then the non-extremal curves C satisfy h1IC (n) ≤ 2. Irene

Sabadini showed [39] that Hd,g has 4 irreducible components for d ≥ 9: (a) extremal

curves, (b) subextremal curves, (c) bilinks of height one from a double line of genus

−2 on a surface of degree d − 2 and (d) disjoint unions of an ACM extremal curve of

degree d − 1 and a line. Families (b) and (c) specialize to (a) by Theorem 9(7) and

family (d) specializes to (a) by Theorem 9(5).

440 S. Nollet

THEOREM 11. For d ≤ 4, Hd,g is connected whenever it is non-empty.

Proof. Since Hd,g is irreducible for g =(d−12

), we may assume that d ≥ 2 and g ≤(

d−22

)by Theorem 6. There are just three cases to consider.

If d = 2, then H2,g consists only of double lines, which were classified by

Migliore [30]. These form an irreducible family.

If d = 3, then H3,g has exactly F 4−g3 G irreducible components, most consistingonly of triple lines. In this case there are curves which lie in the intersection of all the

irreducible components [33, Prop. 3.6 and Remark 3.9], hence H3,g is connected.

Finally if d = 4, then H4,g has roughlyg2

24irreducible components, most of the

families consisting of 4-lines (there are roughly−3g2families whose general member

is not supported on a line). In work of the author and Enrico Schlesinger [36], these

components were classified and connectedness was established through a variety of

methods (see next two sections). One new feature to this example is the existence of an

irreducible component which does not intersect the extremal component: the general

curve is a multiplicity four structure on a line which has generic embedding dimension

three.

Looking at the number of irreducible components of the Hilbert schemes, one

might guess that Hd,g has on the order of gd−2 irreducible components, at least for

g << 0. For degrees d = 2 and d = 3, the reason for the large number of components

is the number of different families of multiplicity structures on a line. Will this behavior

persist for larger d? At the other edge, there are few components for g ∼(d−32

). Can

one find an upper bound on the number?

PROBLEM 8. How many irreducible components does Hd,g have?

(a) For g << 0? Is it of order gd−2? Can one show this is a lower asymptoticbound?

(b) For g near(d−32

)? Can one find an upper bound?

4. The Hilbert scheme H4,−99

In this section we fully describe an example, the Hilbert scheme H4,−99. We list theirreducible components and their dimensions, as well as describing the general curve

in the corresponding family. Complete proofs for general arithmetic genus g can be

found in [36].

REMARK 5. The following refer to Table 1.

(a) Notation: L always denotes a line, D a smooth conic, Z a curve of degree two

with given genus, and W a triple line.

(b) In family G10,m , we set ε(m) = 0 for m > 1, ε(1) = 1 and ε(0) = 3.


(c) Most of the families consist of multiplicity structures on a line.

(1) The thick 4-lines that occur in family G4 are curves C with linear support L

such that IC ⊂ I2L (they contain L(1)).

(2) A multiplicity structure C of degree k on a line L which is not thick is called

quasi-primitive [2] and has a Cohen-Macaulay filtration

L ⊂ Z2 ⊂ Z3 ⊂ · · · ⊂ Zk = C

with quotients IL/IZ2∼= OL(a), IZ2/IZ3

∼= OL(2a + b) and (if necessary)IZ4/IZ3

∼= OL(3a + c) with b ≤ c. The numbers a, b and c give the type ofthe multiple line: thus a double line has type a, a triple line has type (a, b) anda quadruple line has type (a, b, c). We do not give the type for double lines,because the type is determined by the genus.

(d) The last five families listed come with parameters, meaning that there are several

irreducible components. For example, there are actually 32 irreducible families

of curves of G9,a (each consists of a disjoint union of a triple line and a reduced

line), one for each 1 ≤ a ≤ 32. Similarly there are 33 families of type G7,a ,

32 of type G8,a , 50 of type G10,m and 376 of type G11,a,b, for a total of 529

irreducible components.

(e) We prove connectedness by the following plan:

G9,a G10,0/G10,1 G6 G3↘ ↓ ↙ ↓

G8,a → E = G1 ← G2 → G4 ← G11,a,b↗ ↑ ↑

G7,a G5 G11,0,m−1 ⊂ G10,m>1

Each arrow represents a specialization of curves. The extremal component G1draws several arrows. The arrows G6 → G1,G8,a → G1,G9,a → G1 and

G10,0/G10,1 → G1 follow from Theorem 9, parts (5) and (6) and results in [33].

The arrows G2 → G1 and G5 → G1 can be found in [25], as the relevant curves

lie on a double plane. The arrow G7,a → G1 is obtained by actually writing

down equations of the deformation. The arrows G2 → G4 and G3 → G4 arise

by varying a resolution for the Rao module [36, Prop. 4.2 and 4.3], while the

arrow G11,a,b → G4 arises by a tricky deformation of a resolution for the ideals,

using the Buchsbaum-Eisenbud criterion [5] to check exactness [36, Prop. 2.4].

Finally, the curves in G10,m with m > 1 consist of disjoint unions of double

lines of genus < −1. As the support of these curves lies on a smooth quadric,the curves themselves lie on a double quadric. On this surface we were able to

deform these curves to a quasi-primitive 4-line in G11,0,m−1 on a fixed doublequadric: we explain this in the next section. The quasi-primitive 4-lines deform

to G4 as in arrow G11,a,b → G4.

442 S. Nollet

Table 1: The 529 Irreducible Components of H4,−99

Label General Curve Dimension

G1Extremal curves

D ∪ Z

D smooth conic

pa(Z) = −102, length(D ∩ Z) = 4

213

G2Subextremal curves

L1 ∪2P Z ∪2Q L2L1 ∩ L2 = ∅pa(Z) = −101

211

G3

D ∪2P ZD smooth conic

pa(Z) = −100211

G4 thick 4-line 306

G5 double conic 211

G6Z ∪2P L1∪L2pa(Z) = −99 209

G7,a1 ≤ a ≤ 33

W ∪3P LW quasiprimitive 3-line

of type (a, 99− 3a)209− a

G8,a1 ≤ a ≤ 32

W ∪2P LW quasiprimitive 3-line

of type (a, 98− 3a)208− a

G9,a1 ≤ a ≤ 32

W ∪LW quasiprimitive 3-line

of type (a, 96− 3a)206− a

G10,m0 ≤ m ≤ 49

Z1∪Z2pa(Z1) = −m

pa(Z2) = m − 98206+ ε(m)

G11,a,b1 ≤ a ≤ 16

0 ≤ b ≤ 48− 3a

Quasiprimitive 4-line

of type (a, b, c = 96− 6a − b)205− 3a

5. Curves on the double quadric

Hartshorne and Schlesinger gave a satisfying classification of curves lying on the dou-

ble plane [25], describing all the irreducible components and showing connectedness.

Their primary tool was a certain triple associated to such a curve (Definition 1 below).

In this section we describe joint work of Enrico Schlesinger and the author [35], which

uses these triples on a double surface to give a criterion for when the underlying triple

of a curve can be spread out in a flat family. As an application we obtain in Example 3


(a) the inclusion

(3) G11,0,m−1 ⊂ G10,m

needed to show connectedness of H4,−99 (see Remark 5 (e)).

To set the scene, let F be a smooth surface on a smooth threefold T with dou-

bling X = 2F . More generally one can take X to be a ribbon over F in the sense of

Eisenbud and Bayer [3].

DEFINITION 1. For each curve C ⊂ X, the triple T (C) = {Z , R, P} is definedas follows:

1. P is the support of C, the one dimensional part of C ∩ F.

2. R is the curve part of C residual to P.

3. Z is the zero-dimensional part of C ∩ F, so IC∩F,F∼= IZ ,F (−P).

REMARK 6. If T (C) = {Z , R, P}, then Z ⊂ R is zero-dimensional and Goren-

stein [35, Prop. 2.1] and R ⊂ P are divisors on F . The arithmetic genus is given by

(4) pa(C) = pa(P) + pa(R) + degROR(F) − deg Z − 1.

EXAMPLE 2. We show below that both families of curves involved in inclusion

(3) lie on a double quadric in P3 and compute their triples.

(a) A curve C in the family G10,m is a disjoint union C = D1 ∪ D2 of double

lines of genera−m andm−98. The support L1∪L2 being contained in a 3-dimensionalfamily of smooth quadrics, we can choose such a quadric Q containing neither D1 nor

D2. Then C lies on the double quadric X = 2Q and

T (C) = {Z1 ∪ Z2, L1 ∪ L2, L1 ∪ L2}

where Z1 ⊂ L1 has length m + 1 and Z2 ⊂ L2 as length 99− m ≥ m + 1 by formula

(4). For C general, Zi can be taken to be reduced sets of points.

(b) A curve C in the family G11,0,m−1 is a quasi-primitive 4-line supported onL of type (0,m−1, 97−m) (see Remark 5 (c)) and has underlying double line of type0 and hence genus −1. Such a double line necessarily lies on a smooth quadric surfaceQ [33, Remark 1.5], hence C itself lies on the double quadric X = 2Q. It takes some

work [36, Prop. 3.1], but one finds that

T (C) = {Z , 2L , 2L},

where 2L is the double line on Q and Z consists of 98− 2m reduced points and m + 1double points on 2L , none of which are contained in L .

REMARK 7. Looking at the triples in Example 2, we note that triple in part (b)

is a limit of the triples in part (a): The two lines L1 and L2 come together on Q to form

the double line 2L , and the sets of reduced points Z1 and Z2 can be brought together

444 S. Nollet

in this limit to form m + 1 double points and 98− 2m reduced points. If we could liftthis flat family of triples to a flat family of curves on X = 2Q, we would have proved

the inclusion (3).

Thus we consider the map C <→ T (C) = {Z , R, P}, which yields a naturaltransformation of functors

Ht→ D

where H is the set of flat families of curves on X = 2F and D is the set of triples

{Z , R, P}. The functor D is represented by a disjoint union of locally closed sub-

schemes Dz,r,p, where {z, r, p} are the respective Hilbert polynomials of the entries inthe triple {Z , R, P}. The pre-images under t stratify the Hilbert scheme H into locally

closed subschemes Hz,r,p. The map t has a nice structure over the locus of the triples

in D given by a vanishing [35, Thm. 3.2]:

THEOREM 12 (Nollet and Schlesinger, 2003). Let V ⊂ Dz,r,p be the open

subscheme corresponding to triples {Z , R, P} satisfying H1(OR(Z + P − F)) = 0.

Then the map t−1(V ) → V is the composition of an open immersion and an affine

bundle projection. In particular, if Y ⊂ V is irreducible, then t−1(Y ) is also irreducible(hence connected).

EXAMPLE 3. Here are two applications of Theorem 12.

(a) In view of Remark 7, Theorem 12 will prove the inclusion (3) if the vanishing

H1(OR(Z + P − Q)) = 0 holds for both the triples in Example 2. This is easy for the

triples in (a): writing R = L1∪L2 the vanishing boils down to H1(OLi (Zi+1−2)) = 0

for i = 1, 2, which is immediate because deg Zi ≥ 0. The vanishing for family (b)

uses the Cohen-Macaulay filtration (Remark 5 (c)) for the 4-line C [36, Prop. 3.1].

(b) Some of the deformations used in showing the connectedness of H3,g follow

from Theorem 12, for example [33, Prop. 3.3].

We close this section with some open questions involving the fibres of the map

t : H → D. Given a triple T = {Z , R, P} ∈ D on F , the fibre t−1(T ) is the set oflocally Cohen-Macaulay curves C ⊂ X with T (C) = T (there may be none). There is

a bijection between such curves C and surjections φ : IP ⊗OR → OR(Z − F) suchthat φ ◦ τ = σ , where

τ = (O(−F) ↪→ IP ) ⊗OR σ = (OR(−Z) ↪→ OR) ⊗OR(Z − F)

are the natural maps [35, Prop. 2.2], hence these maps can be identified with an open

subset

U ⊂ HomR(OR(−P),OR(Z − F)) ∼= H0(OR(Z + P − F)).

PROBLEM 9. Under what conditions is the open set U non-empty? When does

a given triple T = {Z , R, P} arise from a curve C ⊂ X?

REMARK 8. Obviously a solution to Problem 9 will have applications to clas-

sifying non-reduced curves of low degree. Here are some partial results.


(a) For triple T = {Z , R, P}, the open subsetU is non-empty if any of the followingconditions hold [35, Remark 2.7 and Prop. 2.5]:

(1) H1(OR(Z + P − F)) = 0 and OR(Z + P − F) is generated by globalsections.

(2) H1(OR(Z + P − F − H)) = 0 for a very ample divisor H on R.

(3) H1(OR(P − F)) = 0.

(b) For the double plane X = 2H ⊂ P3, the subset U is non-empty for any triple,

because condition (3) above holds. Chiarli, Greco and Nagel have described the

curves with fixed triple using a matrix of homogeneous polynomials over H ,

giving a certain “normal form” to such curves C [9].

(c) The double quadric X = 2Q ⊂ P3 is more interesting [35, Ex. 2.8]. Let T ={Z , R, P} be a triple with Z Gorenstein of dimension zero.(1) If R = P is a smooth rational curve, then T arises from a curve with one

exception: R = P is a conic and Z is a reduced point.

(2) If P is ample on Q and R /= P , then T arises from a curve.

(3) If R ⊂ P are disjoint unions of rulings on Q, then T arises from a curve if

and only if Z ∩ L /= ∅ for each ruling L ⊂ R.

PROBLEM 10. Answer the question implicit in part (c) above: Which triples on

a smooth quadric in P3 come from a curve on the double quadric? Describe the Hilbertschemes Hd,g(2Q).

PROBLEM 11 (Hartshorne). Which curves on a double surface 2F ⊂ P3 areflat limits of curves on smooth surfaces? For example, the thick triple line L(2) on

the double plane 2H is a flat limit of twisted cubic curves lying on smooth quadric

surfaces. What is special about the curve L(2) or its triple {∅, L , 2L} that allow it to besuch a limit?

6. Subcanonical curves

In view of Paolo Valabrega’s research interests [6, 7, 13, 41], we thought it would

be interesting to determine which families of curves in H4,−99 are subcanonical. Alocal complete intersection curve C is α-subcanonical if ωC ∼= OC (α). The followingrestricts our attention to just a few families in H4,−99.

PROPOSITION 1. Suppose that C ∈ H4,−99 is subcanonical. Then

(1) ωC ∼= OC (−50).

(2) C has no smooth rational irreducible components.

(3) C is one of the following:

446 S. Nollet

(a) A double conic.

(b) A union of two double lines.

(c) A quasi-primitive 4-line.

Proof. An α-subcanonical curve of degree d and genus g satisfies dα = 2g − 2 in

general, hence α = −50 in our case.Suppose that C has a smooth rational component R. Then deg R /= 4 because

then C = R has genus 0 /= −99. Also deg R /= 3 because then C = R ∪ L (L a line)

forces pa(C) = deg(R ∩ L)− 1 ≥ −1 is not equal to −99. Thus R is a line or a conic.We write C = S ∪ R and restrict the exact sequence

0→ ωS ⊕ ωR → ωC → ωS∩R → 0

to R. Using ωC = OC (−50) we obtain

ωS|R ⊕ ωRφ→ OR(−50) → ωS∩R → 0.

Now the sheaf ωS|R is torsion and ωR is either isomorphic ωR = OR(−2) (if R isa line) or OR(−1) (if R is a conic), hence φ is the zero map. This proves (2) by

contradiction, since the cokernel of φ is finitely supported.

Let B = SuppC . Then deg B < 4 (since g < −3) and deg B /= 3 (since then

C consists of a double line and a reduced curve of degree two). Thus deg B = 2 or

1 and C is either (a) a double conic, (b) a union of two double lines or (c) a multiple

line by part (2). If C were a thick 4-line supported on L , then it contains the triple

line with ideal I2L , which has degree 3 and genus 0 (a degenerate twisted cubic curve).

According to [36, Lem. 4.1], C has spectrum

{−98, 0, 12},

which is a shorthand way of saying that the function hC (n) = !2h0OC (n) satisfieshC (−98) = 1, hC (0) = 1, hC (1) = 2 and hC (n) = 0 otherwise. Such a curve

C cannot satisfy ωC = OC (−50), for in this case it would not satisfy the symmetryhC (n) = hC (−50+ 2− n)) [40, Prop. 2.15].

PROPOSITION 2. There are 18 irreducible components of H4,−99 whose gen-eral member is (−50)-subcanonical, as listed in Table 2.

Proof. By Proposition 1 we need only consider (a) double conics, (b) unions of double

lines, and (c) quasi-primitive 4-lines. The double conics are automatically subcanon-

ical, for if D is the support of a double conic C , then the Cohen-Macaulay filtration

is

0→ IC → ID → OD(49) → 0.

Noting that OD(49) = ωD(50), we see that C arises by the Ferrand construction [12]and hence is subcanonical.


Table 2: Irreducible families of subcanonical curves in H4,−99

Label from Table 1 Dimension Spectrum

G5Double conics

211 {−49,−48, 0, 1}

G10,49Disjoint union of two double

lines of genus −49206 {−482, 02}

G11,a,48−3a for 0 < a ≤ 16

Quasi-primitive 4-line205− 3a {−48,−48+ a,−a, 0}

Next consider a union C = D1 ∪ D2 of double lines. If C is connected, then

the support is planar and C is contained in the double plane. It follows that C is a limit

of double conics by Theorem 12 or [25, Thm. 5.1], so we need only consider disjoint

unions of double lines. Since a double line of genus g is (g−1)-subcanonical, a disjointunion of such can only be subcanonical if the double lines have the same genus, which

in this case must be −49.Now let C be a quasi-primitive 4-line of type (a, b, c) with 0 < a ≤ 16, 0 ≤

b ≤ 48− 3a and c = 96− 6a − b. This means that there are locally Cohen-Macaulay

curves L ⊂ D ⊂ W ⊂ C with quotients IL/ID ∼= OL(a), ID/IW ∼= OL(2a + b) andIW /IC ∼= OL(3a + c) (see Remark 5 (c)). Piecing together the exact sequences andusing a > 0, the spectrum of C is

{−3a − c,−2a − b,−a, 0}.

To be (−50)-subcanonical, this sequence of integers must be symmetric about −24[40, Prop. 2.15], which forces b = 48 − 3a and c = 48 − 3a. It now suffices to showthat the general 4-line C of type (a, 48− 3a, 48− 3a) is subcanonical.

The exact sequence

(5) 0→ ID → IL → OL(a) → 0

shows that the underlying double line D ⊂ C arises from the Ferrand construction

and is (−a − 2)-subcanonical, since OL(a) ∼= ωL(a + 2). In fact, D is a divisor on asmooth surface S ⊂ P3 of degree a+2 by [33, Rmk. 1.5]. In view of the isomorphismsIS ∼= OP3(−a−2) and ID,S⊗OD = OS(−D)⊗OD

∼= ωS⊗ω−1D withωS

∼= OS(a−4)and ωD

∼= OD(−a − 2), restricting the exact sequence

0→ IS → ID → ID,S → 0

to D yields

(6) OD(−a − 2) τ→ N∨D

π→ OD(2a − 2) → 0.

448 S. Nollet

Since π is a surjection of bundles on D, the kernel is a line bundle on D. Since any

surjection of line bundles is an isomorphism, τ is injective and sequence (6) is shortexact.

Exact sequence (5) shows that h0OD(m) = h1ID(m) = 0 for m < −a, hencesequence (6) yields the vanishing H1(ND ⊗ ωD(m)) ⊥ H0N∨

D (−m) = 0 for m >3a−2. ThereforeND⊗ωD is (3a)-regular and soND⊗ωD(n) is generated by globalsections for n ≥ 3a by the Castelnuovo-Mumford theorem. Since a ≤ 16, we have in

particular that ND ⊗ ωD(50) is generated by global sections and we obtain a nowherevanishing section yielding a surjection ID → N∨

D → ωD(50) whose kernel IC is theideal sheaf for a (−50)-subcanonical curve C by Ferrand’s construction. Clearly C issupported on L and the sequence

0→ ωD(50) → OC → OD → 0

shows that the spectrum of C is {−48,−48 + a,−a, 0}, so C is quasi-primitive of

type (a, 48 − 3a, 48 − 3a). For curves with fixed spectrum, the property of beingsubcanonical is open and we conclude.

REMARK 9. Chiantini and Valabrega have given equations of such curves [6,

Examples 3.1 and 3.2]. For m, n, u > 0 and p ≥ max{m, n}, they observe that thecurve V with homogeneous ideal

IV = ((xn, ym)u, z p−nxn − w p−m ym = ϕ)

is ((1 − u)p + (m + n)u − 4)-subcanonical. Setting 4 = deg V = mnu, we find just

a few possibilities. When u = 1 we obtain plane curves (m = 4, n = 1) and complete

intersections of two quadrics (m = n = 2). More interesting are these:

(a) m = 2, n = 1 and u = 2. To obtain a (−50)-subcanonical curve we takep = deg ϕ = 52. This is a quasi-primitive 4-line of type (−1, 50, 52). It doesnot appear in Table 1 because such 4-lines are limits of double conics. This one

is a Ferrand doubling of the plane curve with ideal (x, y2).

(b) m = n = 1 and u = 4. To obtain a (−50)-subcanonical curve we take p =deg ϕ = 54. This curve is a quasi-primitive 4-line of type (16, 0, 0).

REMARK 10. Here we make a list of the families of subcanonical curves of

degree four. There are none when g is even. For g = 3 there are plane curves and for

g = 1 there are complete intersections of two quadrics. For odd g < 0 we have:

1. Double conics.

2. Disjoint unions of two lines of genusg+12.

3. Quasi-primitive 4-lines of type (a, −g+3−6a2

, −g+3−6a2

) for 0 < a ≤ −g−36

(this

last family is empty for g > −9, as no such a exist).


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AMS Subject Classification: 14H50

Scott NOLLET, Department of Mathematics, Texas Christian University, TCU Box 298900, Fort Worth,

TX 76129, USA



Syzygy 2005

R.M. Miro-Roig∗ – R. Notari† – M.L. Spreafico†

PROPERTIES OF SOME

ARTINIAN GORENSTEIN RINGS


Abstract. In this paper, we fix a Cohen-Macaulay ideal I ⊂ R = K [x1, . . . , xn ] of dimen-sion 1 and we construct a parameter space G(I, r) for the family of Artinian Gorenstein idealsJ with reg(J ) = r for which I is a tight annihilating ideal. We compute the dimension of

G(I, r) and we prove that if r ; 0 (see section 5 for a precise bound) then all J ∈ G(I, r) area basic double G-link of G on I where G is a suitable Artinian Gorenstein ideal containing

I .

1. Introduction

In recent years many authors focus their attention to study Gorenstein ideals and the

role that they play in various of the applications of Commutative Algebra such as Al-

gebraic Geometry, Algebraic Combinatorics and Number Theory.

It is well known that in codimension 2 Gorenstein ideals and complete intersec-

tion ideals coincide; and in codimension 3 Gorenstein ideals are completely described

from an algebraic point of view by the beautiful structure theorem of Buchsbaum and

Eisenbud which allows one to associate an alternating matrix of odd order to each

Gorenstein ideal of codimension 3. Unfortunately the geometric appearance of Goren-

stein ideals I ⊂ K [x1, . . . , xn] is less understood. For this reason, many authors havegiven geometric constructions of some particular families (cf. [2], [7], [9], [10] among

others). In this paper, we construct a parameter space for the family of Artinian Goren-

stein ideals J with fixed regularity and fixed tight annihilating ideal I and we prove that

if the regularity is big enough then all these Gorenstein ideals J are obtained by basic

double G-link of G on I where G is a suitable Artinian Gorenstein ideal containing I .

Next we outline the structure of the paper. Section 2 provides a brief glossary

of definitions. In section 3 we recall some constructions of Gorenstein ideals and we

point out some features of the constructed ideals. All of them have been successfully

applied in the context of liaison to produce Gorenstein links of given ideals and to

study the Gorenstein liaison classes of some particular ideals. In section 4, we first

introduce the notion of tight annihilating ring R/I for an Artinian Gorenstein ringR/J of arbitrary codimension, given by A. Iarrobino and V. Kanev in [6]. Then weintroduce the new definition of tight resolving ring R/I for an Artinian Gorenstein ringR/J which generalizes the other one in the codimension 3 case. We relate the Hilbertfunction of an Artinian Gorenstein ring R/J to the Hilbert function of a tight resolving

∗Partially supported by MTM 2004-00666.†Members of GNSAGA of INdAM.

451

452 R.M. Miro-Roig – R. Notari – M.L. Spreafico

ring for R/J , and we compare the two notions. We prove that if a Cohen-Macaulay ringR/I of dimension 1 is a tight annihilating or resolving ring for an Artinian Gorensteinring R/J and f ∈ [R]d is a regular form for I then the ideal J ′ = J : f R is an

Artinian Gorenstein ideal, J is a basic double link of J ′ on I if the degree d of f is nottoo large, and R/I is not necessarily either a tight annihilating ring or a tight resolvingring for R/J ′. We end this section giving a numerical criterion to assure that R/I isalso a tight resolving ring for R/J ′.

Section 5 contains the main results of this paper. We fix a Cohen-Macaulay ideal

I ⊂ R of dimension 1 and we construct the parameter space G(I, r) for the ArtinianGorenstein ideals G with reg(G) = r and for which I is a tight annihilating ideal. We

prove that G(I, r) is an open subset of an affine space of dimension deg(R/I ). We alsoconstruct the parameter space BDL(I, sI + d) for the family of Artinian Gorensteinideals L = I + f G with regularity reg(L) = sI + d (sI depends on the geometry of I )

which are basic double G-links of G on I where G is the sum I + I1 of two suitable

directly linked ideals. We prove that BDL(I, sI + d) is an open subset of an affinespace of dimension hR/I (d). The main result of this paper states that if d ≥ sI +1 thenG(I, sI + d) = BDL(I, sI + d).

2. Preliminaries and notation

Let R = K [x1, . . . , xn] be the polynomial ring in the variables x1, . . . , xn over the fieldK , algebraically closed and of characteristic char(K ) = 0. We assume deg(xi ) = 1,for i = 1, . . . , n, and we consider R with the usual induced graduation over Z, i.e.R = ⊕n∈N[R]n, where [R]n contains the homogeneous polynomials of degree n.

DEFINITION 1. Given a homogeneous ideal I ⊆ R, the function

j ∈ Z → hR/I ( j) = dimK [R/I ] j

is the Hilbert function of the ring R/I, where dimK means the dimension as K−vectorspace.

From the definition it follows that, if I /= R, then hR/I (0) = 1.In the following, the dimension of a ring means its Krull dimension.

DEFINITION 2. Let I ⊆ R be a homogeneous ideal. If dim R/I = 0, we saythat I is an Artinian ideal, and R/I is an Artinian ring.

Because of the noetherianity of the ring R, this definition is equivalent to theusual definition (see [1], Ch.6).

Let I ⊆ R be a homogeneous ideal, and let

(1) 0→ Fc → Fc−1 → · · · → F1 → I → 0

be a minimal free resolution of I, with Fi = ⊕nij=1R

βi j (−bi j ).

Properties of some Artinian Gorenstein rings 453

Following the sheaf theory, we define the regularity of an ideal and of its quo-

tient ring.

DEFINITION 3. Let I ⊆ R be a homogeneous ideal. Then the regularity of

R/I is reg(R/I ) = max{bi j − i | i = 1, . . . , c

}while the regularity of I is reg(I ) =

reg(R/I ) + 1.

Now, we define the properties of the ideals we are mainly interested in.

DEFINITION 4. I is a Cohen-Macaulay ideal if c = dim R − dim R/I = n −dim R/I. Equivalently, R/I is a Cohen-Macaulay ring.

DEFINITION 5. I is a Gorenstein ideal if I is a Cohen-Macaulay ideal and

rank(Fc) =∑nc

j=1 βcj = 1, i.e. Fc O R(−tI ) for some integer tI . Equivalently, R/I isa Gorenstein ring.

We recall now the definition of regular element and some well known properties

that we will use in the sequel.

DEFINITION 6. Let I ⊆ R be an ideal with dim R/I = 1, and let f ∈ [R]d .The element f is regular for I if I : f R = I.

PROPOSITION 1. If f ∈ [R]d is a regular element for a Cohen-Macaulay idealI ⊆ R, then the sequence

0→R

I(−d)

f−→

R

I→

R

I + f R→ 0

is exact and hR/I+ f R( j) = hR/I ( j) − hR/I ( j − d).

Now, we collect the properties of the Hilbert function needed later on in the

cases dim R/I = 0, 1.

PROPOSITION 2. Let I ⊆ R be a Cohen-Macaulay homogeneous ideal.

1. If I is Artinian, then hR/I ( j) = 0, for j >> 0.

2. Let R/I be an Artinian Gorenstein ring of regularity sI = reg(R/I ). Let Fc OR(−tI ) be the last module in the minimal free resolution of I. Then

i. sI = tI − n;ii. hR/I ( j) = hR/I (sI − j) for every j ∈ N, and so hR/I (sI ) = 1 and

sI = max{j ∈ N | hR/I ( j) /= 0

}.

The proof follows from [1], Corollaries 4.1.4 and 4.1.6.

The integer sI = reg(R/I ) is also called socle degree of the Artinian Gorensteinring R/I.


PROPOSITION 3. Let I ⊆ R be a homogeneous Cohen-Macaulay ideal of di-

mension 1, and regularity rI = reg(R/I ). Then:

1. hR/I ( j + 1) ≥ hR/I ( j), for j ≥ 0;

2. hR/I (rI ) = hR/I (rI + i) for every i ∈ N.

The integer hR/I (rI ) is called the degree deg(R/I ) of R/I.

The Gorenstein property gives constraints not only on the Hilbert function of a

ring, but also on its minimal free resolution.

In fact, using the graded version of [5], Theorem 1.5, one can prove

PROPOSITION 4. Let I ⊆ R be a Gorenstein ideal. Then, the minimal free

resolution of I is self-dual, i.e.

1. Fc− j O F∗j (−tI );

2. δc− j : Fc− j → Fc− j−1 is equal to δ∗(−tI ) : F∗j (−tI ) → F∗

j+1(−tI )

where F∗ = Hom(F, R) is the dual module of the free module F.

3. Some construction of Artinian Gorenstein rings

In this section we recall some well-known methods to construct homogeneous Artinian

Gorenstein ideals in R and some properties that the corresponding quotient rings have.

The first method is the Buchsbaum-Eisenbud structure Theorem for codimen-

sion 3 graded Gorenstein rings ([5], Theorem 2.1).

THEOREM 1. Let g ≥ 3 be an odd integer, and d1 ≤ · · · ≤ dg be a sequence

of positive integers; set d = 2g−1 (d1 + · · · + dg) and suppose this is an integer, let

ei = d − di , and j = d − 3, and we suppose 1 ≤ d1, dg ≤ j + 1 (so ei ≥ 2).

Let 8 be an alternating g × g matrix with entries from the ring R, such thatthe entry ψi j is homogeneous of degree ei − d j if ei > d j and zero otherwise (so the

entries belong to the maximal ideal of R). Let 8i be the (g − 1) × (g − 1) alternatingmatrix obtained by deleting the i−th row and column of 8. Then the pfaffian Pf(8i ) ishomogeneous of degree di . Let I be the ideal Pf(8) generated by Pf(8i ), i = 1, . . . , g.Then I has grade (height) ≤ 3 in R. If I has grade 3, then I is a graded Gorensteinideal of height 3, and the socle degree of R/I is j = d − 3.

Let λ be the column vector with entries λi = (−1)i Pf(8i ).

i. Suppose I has the maximal possible grade 3. Then I has minimal free resolution

0→ R(−d)λ−→ ⊕g

i=1R(−ei )8−→ ⊕g

i=1R(−di )λT−→ I → 0.

ii. Conversely, if I /= R is a height 3 graded Gorenstein ideal of R, there is analternating matrix 8 as above, such that I = Pf(8).


No generalization of Theorem 1 is known for height ≥ 4 graded Gorenstein

ideals.

A second method of constructing Artinian Gorenstein ideals is the following

(see [4], Ex. 3.2.11):

THEOREM 2. Let ϕ : [R]s → K be a non-degenerate linear map. Let

[I ] j ={

[ker ϕ : (x1, . . . , xn)s− j ] j if j ≤ s

[R] j if j > s.

Then I = ⊕ j∈Z[I ] j is an Artinian Gorenstein ideal of regularity reg(I ) = s + 1.

This construction is equivalent to Macaulay’s inverse system, and allows us to

construct every Artinian Gorenstein ideal with given socle degree. It is a very hard

open problem to relate the linear map ϕ to the minimal free resolution of I or at leastto the Hilbert function of R/I.

The next two methods allow one to construct Gorenstein rings of whatever di-

mension, but we state them only in the Artinian case.

We state the first one as a particular case of [8], Theorem 4.2.1, but it was first

proved in [10].

THEOREM 3. Let I1, I2 ⊆ R be homogeneous Cohen-Macaulay ideals such

that dim R/I1 = dim R/I2 = 1. Assume that J = I1 ∩ I2 is a Gorenstein ideal such

that dim R/J = 1. Then G = I1 + I2 is an Artinian Gorenstein ideal.

REMARK 1. (1) Two ideals I1 and I2 satisfying the hypotheses of the previous

theorem are directly linked, and G is said the sum of directly linked Cohen-Macaulay

ideals.

(2) The Gorenstein ideals arising as sum of two Cohen-Macaulay directly linked ideals

were studied in various papers ([11], [12], [13], for example), and it is known that not

every Gorenstein ideal can be obtained by using that construction (see [11], Example

4.1).

The second and last method is the so-called basic double G-link ([7], Lemma

4.8).

THEOREM 4. Let I ⊆ J ⊆ R be homogeneous ideals such that dim R/I = 1

and dim R/J = 0. Let f ∈ [R]d be a regular form for I. Then it holds:

1. deg(I + f J ) = d deg(I ) + deg(J ).

2. If I is perfect and J is unmixed, then I + f J is unmixed.

3. J/I ∼= (I + f J )/I (d).

4. if R/I and J/I are Cohen-Macaulay and J/I has Cohen-Macaulay type 1 thenJ and I + f J are Artinian Gorenstein ideals.


REMARK 2. In the same hypotheses as above, the ideal I + f J is called basic

double G-link of J on I. It is known that this construction does not give every ArtinianGorenstein ideal (see [2], Example 5.13).

Now, we want to give more details on the Gorenstein ideals arising from Theo-

rems 3 and 4. In particular, we will determine how their resolution looks like.

PROPOSITION 5. In the same notation and hypotheses as Theorem 3, if s =reg(R/J ), then

1. hR/I2( j) = hR/J ( j) + hR/I1(s − j − 1) − deg(R/I1);

2. reg(R/G) = s − 1;

3. hR/G( j) = hR/I1( j) + hR/I1(s − j − 1) − deg(R/I1);

4. if 0→ Fn−1 → · · · → F1 → I1 → 0 is a minimal free resolution of I, then

0→ R(−S) →Fn−1⊕

F∗1 (−S)

→ · · · →F1⊕

F∗n−1(−S)

→ G → 0

is a free resolution of G, not necessarily minimal, where S = s + n − 1.

Proof. Recalling that I2 = J : I1, we compute the first difference of the Hilbertfunctions:

!hR/I2( j) = !hR/J (s − j) − !hR/I1(s − j)

!hR/I2( j − 1) = !hR/J (s − j + 1) − !hR/I1(s − j + 1)

and so on, until

!hR/I2(0) = !hR/J (s − 0) − !hR/I1(s − 0).

By adding all the equations we get:

hR/I2( j) = hR/J (s) − hR/J (s − j − 1) − hR/I1(s) + hR/I1(s − j − 1).

Because of the symmetry of the function !hR/J (!hR/J ( j) = !hR/J (s − j)), theprevious equality can be written as

hR/I2( j) = hR/J ( j) − hR/I1(s) + hR/I1(s − j − 1)

and so the first claim is proved.

The claims (2) and (3) follow from the knowledge of the resolution of G. Then,it is enough to prove the claim (4). The resolution of G can be computed by mapping

cone procedure from the short exact sequence

0→ J → I1 ⊕ I2 → G → 0


that relates all the ideals involved in the construction of G. By using the minimal freeresolutions of I1, J and standard results from liaison theory, it is possible to compute afree resolution of I2. From that last one, we get the claim on the free resolution of G.Also if the resolution is non minimal, it is not possible to cancel the last from the left

free module in its resolution, because G is Artinian, and so we get also the result on

the regularity of R/G.

PROPOSITION 6. In the same notation and hypotheses of Theorem 4, if d =deg( f ) then

1. hR/I+ f J ( j) = hR/I ( j) + hR/J ( j − d) − hR/I ( j − d);

2. if 0→ Fn−1 → · · · → F1 → I → 0 is a minimal free resolution of I, then

0→ R(−S) →Fn−1⊕

F∗1 (−S)

→ · · · →F1⊕

F∗n−1(−S)

→ J → 0

is a free resolution of J, not necessarily minimal, for some integer S;

3. reg(R/I + f J ) = reg(R/J ) + d.

Proof. At first, we have the following equality: I ∩ f J = f I.The inclusion ⊇ is evident. The inverse inclusion is an easy consequence of the regu-

larity of f for I. In fact, if f g ∈ I for some g ∈ J then g ∈ I : f R = I and hence

f g ∈ f I.

Now, we have that the short sequence

0→ I (−d) → I ⊕ J (−d) → I + f J → 0

is exact, and so we get the claim on the Hilbert function of R/I + f J .

From the proof of Lemma 4.8 in [7] we know that the two sequences

0→ I → J → J/I → 0

and

0→ I → I + f J → J/I (−d) → 0

are exact, with both J and I + f J Artinian Gorenstein ideals.

We can choose d sufficiently large so that the generators of I + f J not in I

have degree at least 2+ reg(R/I ). Then, we can apply Theorem 3.2 in [3] and we getthat a minimal free resolution of J/I (−d) is

0→ R(−s) → F∗1 (−s) → · · · → F∗

n−1(−s) → J/I (−d) → 0

for some integer s. Hence, the claim on the free resolution of J follows, too.


4. A class of Artinian Gorenstein ideals

One of the most studied class of Gorenstein rings of dimension d ≥ 0 is the class of

quotients of Cohen-Macaulay rings of dimension d+1.Geometrically, they correspondto divisors on arithmetically Cohen-Macaulay projective schemes. In particular, it is

very interesting the case when the two rings have the same Hilbert function in small

degrees, and hence we recall the definition of tight annihilating ring given by Iarrobino

and Kanev ([6], Definition 5.1), which describes that situation.

DEFINITION 7. Let R/J be an Artinian Gorenstein ring. Let I ⊆ J be a

homogeneous ideal such that R/I is a dimension 1 Cohen-Macaulay ring. We say thatR/I is a tight annihilating ring for R/J if hR/J ( j) = hR/I ( j) for j ≤ reg(R/I ), andhR/J ( j) ≤ hR/I (reg(R/I )) = deg(R/I ), for every j ∈ Z.

REMARK 3. We know that the Hilbert function of the Artinian Gorenstein ring

R/J is symmetric, while the one of the ring R/I is increasing until it reaches its max-imum value deg(R/I ). Then, if I is tight annihilating for J, the Hilbert function ofR/J increases as the one of R/I, reaches the value deg(R/I ), and after that, it takesthe same value for some integers, and when it takes a different value, it can be com-

pleted by symmetry. Then, reg(R/J ) ≥ 2 reg(R/I ).

In codimension 3, we can characterize the minimal free resolution of an Artinian

Gorenstein ideal J having a tight annihilating ideal I. In fact, it holds:

PROPOSITION 7. Let I ⊆ R = K [x, y, z] be a Cohen-Macaulay homogeneousideal such that dim R/I = 1 and hR/I (1) = 3. Let J ⊇ I be an Artinian Gorenstein

ideal for which I is tight annihilating. If 0 → F2 → F1 → I → 0 is a minimal free

resolution of I then there exists an integer tJ ≥ 3+ 2 reg(R/I ) such that

0→ R(−tJ ) →F2⊕

F∗1 (−tJ )

→F1⊕

F∗2 (−tJ )

→ J → 0

is the minimal free resolution of J.

Proof. See [6], Theorems 5.31, 5.39, 5.46, and Remark 5.43. The proof is based on

the Buchsbaum-Eisenbud structure theorem for codimension 3 Gorenstein ideals.

This property is shared also from the ideals arising from Theorems 3 and 4.

Then, we choose this last property for defining the tight resolved ring for an Artinian

Gorenstein ring R/J of whatever codimension.

DEFINITION 8. Let R/I be a graded Cohen-Macaulay ring of dimension 1 withminimal free resolution

0→ Fn−1 → Fn−2 → · · · → F1 → R → R/I → 0.


We say that R/I is a tight resolving ring for the Artinian Gorenstein graded ring R/J(or I is a tight resolving ideal for J ) if the minimal free resolution of R/J is

0→ R(−tJ ) →Fn−1⊕

F∗1 (−tJ )

→ · · · →F1⊕

F∗n−1(−tJ )

→ R → R/J → 0

for some integer tJ ≥ n + 2 reg(R/I ).

PROPOSITION 8. Let R/I be a tight resolving ring for the Artinian Gorensteingraded ring R/J. Then, the Hilbert function of R/J is equal to

hR/J ( j) = hR/I ( j) + hR/I (reg(R/J ) − j) − deg(R/I ).

Proof. From the additivity of the Hilbert function on exact sequences, we have

hR/I ( j) =∑n−1

i=0 (−1)i dimK [Fi ] j , where F0 = R, and, if Fi = ⊕nih=1R(−bih)βih ,

then dimK [Fi ] j =∑ni

h=1(j+n−1−bih

n−1). Of course,

hR/J ( j) =n∑

i=0

(−1)i[dimK [Fi ] j + dimK [F∗

n−i ] j−tJ]

=

=n−1∑

i=0

(−1)i dimK [Fi ] j +n∑

i=1

(−1)i dimK [F∗n−i ] j−tJ .

The sequence

0→ R(−tJ ) → F∗1 (−tJ ) → · · · → F∗

n−1(−tJ )

is a free resolution of the canonical module of R/I and so

n∑

i=1

(−1)i dimK [F∗n−i ] j−tJ = dimK

[Extn−1(R/I, R)

]

j−tJ.

It is well known that

dimK

[Extn−1(R/I, R)

]

j−tJ= dimK

[Extn−2(I, R(−n))

]

j+n−tJ,

and by Serre’s duality, we have that

dimK

[Extn−2(I, R(−n))

]

j+n−tJ= h1(Pn−1, I(tJ − n − j))

where I is the ideal sheaf obtained by sheaffifing the saturated ideal I.Hence, the claimfollows.

REMARK 4. Because of the description of the Hilbert function of a dimension

1 Cohen-Macaulay graded ring R/I, we known that hR/I (k) ≤ deg(R/I ) for everyk ∈ Z. Hence, hR/J ( j) ≤ deg(R/I ), for every j ∈ Z.


Now, we compare the notion of tight annihilating and tight resolving ideal for

an Artinian Gorenstein ideal J.

PROPOSITION 9. Let J ⊆ R be an Artinian Gorenstein ideal and let I ⊆ J be

a tight resolving ideal for J. Then, I is a tight annihilating ideal for J if, and only if,reg(R/J ) ≥ 2 reg(R/I ).

Proof. We proved in Proposition 8 above that

hR/J ( j) = hR/I ( j) + hR/I (reg(R/J ) − j) − deg(R/I ).

It follows that hR/J ( j) = hR/I ( j) for every j ≤ reg(R/I ) if, and only if, reg(R/J ) −j ≥ reg(R/I ) for each j ≤ reg(R/I ), i.e. reg(R/J ) ≥ 2 reg(R/I ).

On the other hand, in the codimension 3 case, we have that if an ideal is tight

annihilating for an Artinian Gorenstein ideal J then it is tight resolving for J, too, asexplained in Proposition 7. Because of the absence of a structure theorem for Goren-

stein ideals in codimension ≥ 4, the best we can say is the following:

PROPOSITION 10. Let J ⊆ R be an Artinian Gorenstein ideal and let I ⊆ J

be a tight annihilating ideal for J. Then, if the degrees of the minimal generators of Jnot in I are at least 2+ reg(R/I ), then I is a tight resolving ideal for J.

Proof. The claim is [3], Theorem 3.2.

Now, we want to construct new Artinian Gorenstein ideals from a given one.

THEOREM 5. Let I ⊆ J be homogeneous ideals in R. Assume that R/J is anArtinian Gorenstein ring and that R/I is a dimension 1 Cohen-Macaulay ring. Let dbe an integer such that hR/J (d) = hR/I (d), and let f ∈ [R]d be a regular form for I.Then, J ′ = J : f R is an Artinian Gorenstein ideal.

Proof. At first, we prove that J ′ is an Artinian ideal. In fact,

hR/J (reg(R/J ) + 1) = hR/J (reg(R/J ) + i) = 0 for every i ≥ 1,

and this is equivalent to the equality [J ] j = [R] j for j ≥ reg(R/J ) + 1.

If g ∈ [R] j−d , with j ≥ reg(R/J ) + 1, then g f ∈ [R] j = [J ] j , and hence g ∈[J : f R] j−d = [J ′] j−d , and this proves that [J ′] j = [R] j for each j ≥ reg(R/J ) −d + 1, i.e. J ′ is an Artinian ideal.

Now, we prove that J ′ is a Gorenstein ideal.

Let ϕ : [R]reg(R/J ) → K be a non-degenerate K−linear map such that ker ϕ =[J ]reg(R/J ). We define ψ : [R]reg(R/J )−d → K to be the K−linear map such thatψ(g) = ϕ(g f ). According to Theorem 2, we prove that ψ is non degenerate and that

[kerψ : (x1, . . . , xn)j ]reg(R/J )−d− j = [J ′]reg(R/J )−d− j .


If ψ(g) = 0 for every g ∈ [R]reg(R/J )−d , then ϕ(g f ) = 0 for every g ∈[R]reg(R/J )−d i.e. f ∈ [ker ϕ : (x1, . . . , xn)

reg(R/J )−d ]d = [J ]d , with d ≤ reg(R/J )−reg(R/I ). But hR/J (d) = hR/I (d) and so [J ]d = [I ]d . Hence, f ∈ [I ]d and so f is

not regular for I. This contradiction proves that ψ is non degenerate.

Now, g ∈ [kerψ : (x1, . . . , xn)j ]reg(R/J )−d− j if and only if ϕ(g f h) = 0 for

every h ∈ (x1, . . . , xn)j . By definition, this means that g f ∈ [J ]reg(R/J )− j i.e. g ∈

[J : f R]reg(R/J )−d− j = [J ′]reg(R/J )−d− j .

Conversely, if g ∈ [J ′]reg(R/J )−d− j , then g f ∈ ker ϕ : (x1, . . . , xn)j and

deg(g f ) = reg(R/J ) − j, i.e. ϕ(g f h) = 0,∀h ∈ (x1, . . . , xn)j . From the defini-

tion of ψ, it follows that ψ(gh) = 0 for each h ∈ (x1, . . . , xn)j and so g ∈ [kerψ :

(x1, . . . , xn)j ]reg(R/J )−d− j .

REMARK 5. Let I, J, J ′ be as above. Then I ⊆ J ⊆ J ′.

REMARK 6. If 1 ≤ d ≤ reg(R/J ) − reg(R/I ) then hR/J (d) = hR/I (d) bothin the case I is a tight annihilating ideal for J and in the case I is a tight resolving ideal

for J.

Now, we give an example to show that the Hilbert function of R/J ′ depends onJ and f and not only on J and d = deg( f ).

EXAMPLE 1. Let I ⊆ R = K [x, y, z] be the ideal generated by y3− xz2, x3−y2z, z3 − x2y. Its minimal free resolution is

0→R(−4)⊕

R(−5)

A−→ R3(−3) −→ I → 0

where

A =

z x2

y z2

x y2

.

The ideal I1 = (x, y2) is geometrically linked to I via the complete intersection ideal(y3 − xz2, x3 − y2z), and the forms f = x6 + y6 + z6 and g = x5y + y5z + xz5 are

regular for I.

Hence, J = I + f I1 is an Artinian Gorenstein ideal with Hilbert function

hR/J = (1, 3, 6, 7, 7, 7, 7, 6, 3, 1).

The ideals J1 = J : f R = (x, y2, z3) and J2 = J : gR = (x2 − xy − z2, xy −xz− yz, y2−3xz−2yz− z2) are Artinian Gorenstein with different Hilbert functions.In fact, we have

hR/J1 = (1, 2, 2, 1)

while

hR/J2 = (1, 3, 3, 1)


and so the degree of the regular form is not enough to compute the Hilbert function of

the new Artinian Gorenstein ideal.

However, if the degree d of f is not too large, then the Hilbert function of J ′

depends only on J and d. In fact, it holds:

PROPOSITION 11. In the same hypotheses of Theorem 5, assume furthermore

that I is a tight annihilating (resp. tight resolving) ideal for J. If d ≤ reg(R/J ) −2 reg(R/I ) + 1, then

hR/J ′( j) = hR/I ( j) + hR/I (reg(R/J ′) − j) − deg(R/I ).

Proof. We know that J ⊆ J ′ and so [J ]k ⊆ [J ′]k for every integer k.Let g ∈ [J ′]k . By construction, g f ∈ [J ]k+d . If k + d ≤ reg(R/J )− reg(R/I )

then [J ]k+d = [I ]k+d by previous Remark 6, and so g ∈ [I ]k because f is regular

for I. Hence, [J ]k = [J ′]k for every k such that k + d ≤ reg(R/J ) − reg(R/I ) i.e.k ≤ reg(R/J ) − reg(R/I ) − d.We proved that J ′ is an Artinian Gorenstein ideal andso its Hilbert function is symmetric. Then, the Hilbert function of R/J ′ is completelydetermined if the condition k ≤ reg(R/J )−reg(R/I )−d covers at least the first half ofthe range where the Hilbert function is non zero. The largest d for which that happens

is d = reg(R/J ) − 2 reg(R/I ) + 1 and the claim follows.

REMARK 7. If d = reg( RJ) − 2 reg( R

I) + 1, then reg( R

J ′ ) = 2 reg( RI) − 1 and

the Hilbert function of R/J ′ is the one of R/J after erasing its flat part, that is to say,the values where it reaches deg(R/I ).

We will show how the ideals J, I and J ′ are related. To this aim, we need someproperties which we collect in the following lemma.

LEMMA 1. Let I, J, J ′ and f ∈ [R]d be as in Theorem 5. Then:

(1) I ∩ f J ′ = f I ;

(2) J/I : f R/I = J ′/I.

Proof. 1) The inclusion I ∩ f J ′ ⊇ f I follows from the fact that I ⊆ J ′.

Let g ∈ I be an element such that g = f h, h ∈ J ′. Then, h ∈ I : f R = I and

so g ∈ f I, and the other inclusion is verified.

2) It is evident that I ⊆ J ′, because I ⊆ J ⊆ J ′. Then, we can consider theideals J/I, J ′/I, and f R/I of R/I, where f is the class of f in R/I. We want toprove that J/I : f R/I = J ′/I.

Now, if g ∈ J ′/I then g + h ∈ J ′ for some h ∈ I. But I ⊆ J ′, and so g =(g+h)−h ∈ J ′. By its definition, g f ∈ J and so g f ∈ J/I. Hence, g ∈ J/I : f R/I.

Conversely, if g ∈ J/I : f R/I, then g f ∈ J/I. Of course, there exists h ∈ I

such that g f + h ∈ J. As for the reverse inclusion, from I ⊆ J we get that g f =(g f + h) − h ∈ J. By its definition, it holds that g ∈ J : f R = J ′, i.e. g ∈ J ′/I.


Now, we can prove that, if d is not too large, then J is a basic double link of J ′

on I.

PROPOSITION 12. Let I, J, J ′ and f ∈ [R]d be as in Theorem 5. More-

over assume that I is either tight annihilating or tight resolving for J. Then, if d ≤reg(R/J ) − 2 reg(R/I ) + 1, then J = I + f J ′.

Proof. We know that f J ′ ⊆ J and so I + f J ′ ⊆ J. Then, the equality follows ifthey have the same Hilbert function. The Hilbert function of I + f J ′ was computed inProposition 6(1) and it is

hR/I+ f J ′( j) = hR/I ( j) + hR/J ′( j − d) − hR/I ( j − d).

By Proposition 11, we have

hR/I+ f J ′( j) = hR/I ( j) + hR/I (reg(R/J ) − j) − deg(R/I ) = hR/J ( j)

and the claim follows.

Now, we show with an example that R/I could be neither a tight annihilatingring nor a tight resolving ideal for R/J ′.

EXAMPLE 2. Let R = K [x, y, z] be a polynomial ring in 3 unknowns, and letI ⊆ R be the ideal generated by y2 − xz, x2 − yz, z2 − xy whose resolution is

0→ R2(−3) A−→ R3(−2) → I → 0

where

A =

z x

y z

x y

.

The Hilbert function of R/I is hR/I = (1, 3,→), and so reg(R/I ) = 1.

Let J = (y2 − xz, x2 − yz, z2 − xy, xz,−xy) be an Artinian Gorenstein ideal,whose minimal free resolution is

0→ R(−5) → R5(−3) B−→ R5(−2) → J → 0

where

B =

0 0 −x z x

0 0 0 y z

x 0 0 x y

−z −y −x 0 0

−x −z −y 0 0

.

We have I ⊆ J, and tJ = 5, reg(R/J ) = 2. It is evident that R/I is a tight annihilatingring for R/J.We can control the Hilbert function of R/J ′ for every d ≤ 1.

The element x ∈ [R]1 is general for I ; in fact, I : x R = I.Moreover J : x R =(y, z, x2) = J ′ and R/I is not a tight annihilating ring for R/J ′, because the Hilbert


function of R/J ′ verifies hR/J ′(1) = 1 /= hR/I (1) = 3. Moreover, the minimal freeresolution of R/J ′ is

0→ R(−4) →R(−2)⊕

R2(−3)→

R2(−1)⊕

R(−2)→ R → R/J ′ → 0

and it does not contain the minimal free resolution of R/I as a subcomplex, and henceR/I is not a tight resolving ideal for R/J ′.

Now, we compute the shape of a free resolution of R/J ′.

LEMMA 2. In the same hypotheses as above, the shape of a free resolution of

R/J ′ is

0→ R(−tJ ′) →Fn−1⊕

F∗1 (−tJ ′)

→ · · · →F1⊕

F∗n−1(−tJ ′)

→ R → R/J ′ → 0

where tJ ′ = tJ − d.

Proof. From the assumptions on the minimal free resolutions of R/I and R/J it fol-lows the diagram

0 → Fn−1δn−1−→ · → F1 → R → R/I → 0

↓ ↓ ↓ ↓

0→ R(−tJ ) →Fn−1⊕

F∗1

(−tJ )→ · →

F1⊕

F∗n−1(−tJ )

→ R → R/J → 0

↓ ↓ ↓ ↓ ↓0→ R(−tJ ) → F∗

1(−tJ ) → · → F∗

n−1(−tJ ) → coker(δ∗n−1) → 0

with exact rows and split exact columns.

Hence, 0→ coker(δ∗n−1)(−tJ ) → R/I → R/J → 0 is a short exact sequence.

It follows that J/I O coker(δ∗n−1)(−tJ ).But, we know that J/I O f (J ′/I ) and that f : R/I (−d) → R/I is an injective

map. It follows that J ′/I O coker(δ∗n−1)(−tJ + d) = coker(δ∗n−1)(−tJ ′). A free

resolution of J ′/I is then

0→ R(−tJ ′) → F∗1 (−tJ ′) → · · · → F∗

n−1(−tJ ′) → coker(δ∗n−1)(−tJ ′) → 0

and so

0→ R(−tJ ′) →Fn−1⊕

F∗1 (−tJ ′)

→ · · · →F1⊕

F∗n−1(−tJ ′)

→ R → R/J ′ → 0

is a (eventually non minimal) free resolution of R/J ′.

Now, we give a numerical criterion to guarantee that I is a tight resolving ideal

for J ′, too.


PROPOSITION 13. Let I, J, J ′ be as above. If d ≤ reg(R/J )−2 reg(R/I ) thenR/I is a tight resolving ring for R/J ′.

Proof. At first, we check that hR/J ′( j) = hR/I ( j) for j ≤ reg(R/I ).

Thanks to Proposition 11, we verify that hR/I (tJ ′ − j − n) = deg(R/I ) forevery j ≤ reg(R/I ), i.e. that tJ − d − n − j ≥ reg(R/I ) for every j ≤ reg(R/I ). Byhypothesis, reg(R/J )− d ≥ 2 reg(R/I ), and so the previous inequality can be writtenas 2 reg(R/I ) − j ≥ reg(R/I ) that holds for every j ≤ reg(R/I ).

Now, we want to check that the resolution of R/J ′ we computed is minimal.The resolution is obtained by mapping cone, and so we have to check that no en-

try of a matrix representing F∗n− j−1(−tJ ) → Fj has degree zero. Using the nota-

tion stated in Section 2, we have that Fj = ⊕n jk=1R

β jk (−b jk) and F∗n− j−1(−tJ ) =

⊕nn− j−1h=1 Rβn− j−1,h (−tJ+d−bn− j−1,h), and so we have to prove that tJ−d−bn− j−1,h−

b jk > 0 for every h and k. By substituting the regularity of R/J we get

tJ − d − bn− j−1,h − b jk = n + reg(R/J ) − d − bn− j−1,h − b jk ≥≥ n + 2 reg(R/I ) − bn− j−1,h − b jk ≥≥ n − (n − j − 1) − j = 1

(2)

where the first inequality follows from our hypothesis on d and the second one from

the definition of regularity. Then, no cancellation can be performed and the resolution

is minimal.

REMARK 8. The regularity reg(R/J ′) of R/J ′ is equal to reg(R/J ′) =reg(R/J ) − d. If d ≤ reg(R/J ) − 2 reg(R/I ) then reg(R/J ′) ≥ 2 reg(R/I ) andso R/I is a tight annihilating ring for R/J ′ by Proposition 9.

Before ending the section, we compute the residual of J with respect to J ′.

PROPOSITION 14. Let J be an Artinian Gorenstein ideal, let I ⊆ J be a tight

resolving ideal for J and let J ′ = J : f R where f ∈ [R]d is a regular form for I, withd ≤ reg(R/J ) − reg(R/I ). Then, J : J ′ = J + f R.

Proof. The inclusion J ⊆ J ′, observed in Remark 5, induces a map of complexesbetween the minimal free resolution of J and J ′:

0 → R(−tJ ) → Hn−1 → . . . → H1 → J → 0

↓ ↓ ↓ ↓0 → R(−tJ ′) → Gn−1 → . . . → G1 → J ′ → 0

and so we get the equality J : J ′ = J + gR where g ∈ [R]tJ−tJ ′=d represents the lastmap R(−tJ ) → R(−tJ ′) (see [2]). By the definition of J ′ we get f J ′ ⊆ J. Hence,f ∈ J : J ′ = J + gR.

If f ∈ J, then f ∈ I, because of its degree, and this is not possible being f

regular for I. It follows that f /∈ J and so f = ag + h, a ∈ K − {0} , h ∈ J, i.e.J : J ′ = J + f R.


5. The main result

In this section, chosen a dimension 1 Cohen-Macaulay ring R/I that contains an idealG/I such that R/G is an Artinian Gorenstein ring, we want to compare the ideals

which are basic double G-links of G on I with the Artinian Gorenstein ideals contain-

ing I for which I is a tight annihilating ideal.

With this in mind, we construct an Artinian Gorenstein ideal G as the sum of I

and an ideal I1 which is geometrically linked to I (see Theorem 3).

CONSTRUCTION 1. Let J be a Gorenstein ideal that verifies the following con-

ditions

1. J ⊆ I ;

2. dim R/J = 1;

3. if I1 = J : I then J = I ∩ I1;

4. the minimal generators of J can be choosen among the minimal ones of I.

Then, the ideal G = I + I1 is an Artinian Gorenstein ideal, which is the sum of two

directly linked Cohen-Macaulay ideals.

Of course, there are dimension 1 Cohen-Macaulay rings for which the construc-

tion does not work, and that depends on the geometry of the schemes defined by those

rings, as the following example shows.

EXAMPLE 3. Let P2 = Proj(K [x, y, z]), and let X,Y, Z be three 0-dimen-sional schemes of degree 11 defined by the ideals IX = (x3 − y2z, z4 − xy3, y5 −x2z3), IY = (x3,−xy3, y5 − x2z3) and IZ = (x3,−xy3, y5), respectively. The threeschemes have the same Hilbert function

hX = hY = hZ = (1, 3, 6, 9, 11,→).

The minimal free resolution of IX is

0→ R2(−6) A−→ R(−3) ⊕ R(−4) ⊕ R(−5) → IX → 0

where

A =

y3 z3

x2 y2

z x

.

The first two generators of IX form a regular sequence, and so J = (x3−y2z, z4−xy3)is a complete intersection ideal. The ideal I1 = J1 : IX = (x, z) has degree 1 andJ1 = IX ∩ I1.

The minimal free resolution of IY is

0→ R2(−6) B−→ R(−3) ⊕ R(−4) ⊕ R(−5) → IY → 0


where

B =

y3 z3

x2 y2

0 x

.

It is evident that, if F ∈ [IY ]4, then x3, F is not a regular sequence because (x3, F) ⊆(x). Hence, the minimal degrees of two generators of IY that form a regular sequenceare 3, 5, and the corresponding ideal J has degree 15.

The geometrical reason for that behavior is that Y contains a degree 5 sub-

scheme Y ′ contained in a line: IY ′ = (x, y5) ⊇ IY .

The minimal free resolution of IZ is

0→ R2(−6) C−→ R(−3) ⊕ R(−4) ⊕ R(−5) → IZ → 0

where

C =

y3 0

x2 y2

0 x

.

Z is supported on the point A(0 : 0 : 1) with IA = (x, y). If J1 ⊆ IZ is a complete

intersection with generators of degrees 3, 5, then J1 = (x3, xy3l + y5) for some l ∈[R]1.

We have J1 : IZ = (x2, y2+xl) = I1 and IZ ∩ I1 = (x3, y3(y2+xl), xy3) ⊂ J

(in fact, xy3 = xy(y2+ xl)− yl(x2)) and hence no complete intersection ideal J givesa geometric link of Z with another scheme. This happens because Z is not locally

Gorenstein.

REMARK 9. We were informed by A. Iarrobino that M. Boij, in a talk at North-

eastern University, proved that there exists an Artinian Gorenstein ideal J ⊇ I with

I tight annihilating for J if, and only if, the ring R/I is locally Gorenstein, i.e. everylocalization of R/I at a minimal prime is a Gorenstein ring.

It is natural to look for an ideal J of minimal socle degree to construct the ideal

I1. Then, we define

DEFINITION 9. Let I be a Cohen-Macaulay ideal such that R/I has dimension1 and is locally Gorenstein. We set

sI = min {reg(R/J )|J verifies the hypotheses of Construction 1} .

If we write a minimal free resolution of an ideal J of minimal socle degree we

have

0→ R(−sI − n + 1) → Qn−2 → · · · → Q1 → J → 0.

As we showed in Example 3, the integer sI depends on the geometry of the ring

R/I.


Let J be a Gorenstein ideal fulfilling all the assumptions of Construction 1, and

verifying reg(R/J ) = sI . Then, we fix now and forever, the ideal G = I + I1, whereI1 = J : I. A free resolution of G was computed in Proposition 6(2).

Now, we want to construct a parameter space for the family of the Artinian

Gorenstein ideals obtained by basic double G-link from G on I, that are I + f G =I + f I1, for some f ∈ R regular for I.

The key property to construct this parameter space is the following:

PROPOSITION 15. Let f1, f2 ∈ [R]d be two elements, regular for I. Then I +f1 I1 = I + f2 I1 if, and only if, f1 = f2 mod (I ).

Proof. First, assume f1 = f2 mod (I ). Let g ∈ I + f2 I1 be a form. Then, there exist

p ∈ I and q ∈ I1 such that g = p + f2q. By assumption, there exists h ∈ I such that

f2 = f1 + h. Hence, g = p + f1q + hq = (p + hq) + f1q = p1 + f1q ∈ I + f1 I1because p1 = p + hq ∈ I.

Vice versa, if L1 = I + f1 I1, L2 = I + f2 I1 and L1 = L2 then, L1 : G = L2 :G. By Proposition 14, I + f1R = I + f2R and then f1 − f2 ∈ I.

We are able to construct the parameter space for the family of the Artinian

Gorenstein ideals which are basic double G-links of G on I.

THEOREM 6. Let I ⊆ R be a dimension 1, locally Gorenstein, Cohen-

Macaulay ideal. Let

BDL(I, sI + d) = {L = I + f I1| deg( f ) = d, f regular for I }

be the family of the Artinian Gorenstein ideals that are basic double G-links of G on

I, of regularity reg(L) = sI + d. Then, BDL(I, sI + d) is parametrized by an opensubset of an affine space of dimension hR/I (d).

Proof. Every ideal L ∈ BDL(I, sI + d) corresponds to the choice of f ∈ [R]d suchthat I : f R = I, that is an open condition.

By Proposition 15, the Artinian Gorenstein ideals L1 = I + f1 I1 and L2 =I + f2 I1 are equal if, and only if, f1 − f2 ∈ I.

Hence, the natural parameter space for BDL(I, sI + d) is the open subset W of

the affine space [R/I ]d corresponding to the forms that are regular for I.By definition, dimW = dimK [R/I ]d = hR/I (d).

Now, following [6], we construct the parameter space for the Gorenstein ideals

for which I is a tight annihilating ideal.


Macaulay ideal. Let

G(I, r) = {L|I is a tight annihilating ideal for L , reg(L) = r}


be the family of Artinian Gorenstein ideals L containing I as tight annihilating ideal,

of regularity reg(L) = r. Then, the parameter space of G(I, r) is an open subset of anaffine space of dimension deg(R/I ).

Proof. The statement in the case of codimension 3 is part of the more general Theorem

5.31 in [6]. But the proof holds verbatim in our hypotheses.

Now, we can state our main result.


Macaulay ideal. Let d ≥ sI + 1 be an integer. Then G(I, sI + d) = BDL(I, sI + d).

Proof. If d ≥ sI +1, then hR/L( j) = hR/I ( j) ∀ j ≤ sI .Moreover, by using argumentsas in the proof of Proposition 13, we can prove that a minimal free resolution of L is

0→ R(−t) →Fn−1⊕

F∗1 (−t)

→ · · · →F1⊕

F∗n−1(−t)

→ L → 0,

where t = sI + n + d − 1, and so I is a tight annihilating ideal for every L ∈BDL(I, sI+d).Hence, BDL(I, sI+d) ⊆ G(I, sI+d).Moreover, bothBDL(I, sI+d)and G(I, sI + d) are parametrized by open subsets of affine spaces of dimensiondeg(R/I ) = hR/I (d). Now, to prove that BDL(I, sI + d) = G(I, sI + d), it is enoughto prove that BDL(I, sI + d) is closed in G(I, sI + d).

If L ∈ G(I, sI + d), and it is the flat limit of a 1-parameter flat family of idealsin BDL(I, sI + d), then there exists a 1-parameter family of polynomials ft ∈ [R]dsuch that Lt = I + ft I1 → L for t → 0.

It is evident that I + ft I1 → I + f0 I1 = L , for t → 0, with f0 ∈ [R]d .If f0 is not regular for I, then there exists a minimal homogeneous prime ideal

P ∈ R/I that contains f0R + I, and so dim RI+ f0 I1

/= 0. But this is a contradictionbecause L is Artinian, and so f0 is regular for I.

Hence, L is a basic double G-link of G on I, and so we get the claim.

The same argument as above proves also the following

PROPOSITION 16. Let I, J be as above, and let 1 ≤ d ≤ sI be an integer.

Then BDL(I, sI + d) is a quasi projective subscheme of G(I, sI + d) of codimensiondeg(R/I ) − hR/I (d).

In R = K [x, y, z], it is known that the first half of the Hilbert function of the Ar-tinian Gorenstein ring R/L is admissible as Hilbert function of a dimension 1 Cohen-Macaulay ring R/I.Moreover, if hR/L( j) = s for at least 3 consecutive integers, then

there exists a dimension 1 Cohen-Macaulay ring R/I of degree deg(R/I ) = s that is

a tight annihilating ring for L . In higher codimension, we could not find an analogous

numerical condition to guarantee the existence of a tight annihilating ring for L .


References

[1] ATIYAH M.F. AND MACDONALD I.G., Introduction to commutative algebra, Addison-Wesley Pub-

lishing Company, Reading, MA 1969.

[2] BOCCI C., DALZOTTO G., NOTARI R. AND SPREAFICO M.L., An iterative construction of Goren-

stein ideals, Trans. Amer. Math. Soc. 357 (4) (2005), 1417–1444.

[3] BOIJ M.,Gorenstein Artin algebras and points in projective space, Bull. LondonMath. Soc. 31 (1999),

11–16.

[4] BRUNS W. AND HERZOG J., Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics

39, Cambridge University Press, Cambridge 1993.

[5] BUCHSBAUM D.A. AND EISENBUD D., Algebra structures for finite free resolutions, and some struc-

ture theorems for ideals of codimension 3, Amer. J. Math. 99 (3) (1977), 447–485.

[6] IARROBINO A.V. AND KANEV V., Power sums, Gorenstein algebras, and determinantal loci, LNM

1721, Springer 1999.

[7] KLEPPE J.O., MIGLIORE J.C., MIRO-ROIG R.M., NAGEL U. AND PETERSON C., Gorenstein li-

aison, complete intersection liaison invariants and unobstructedness, Mem. Amer. Math. Soc. 154,

Amer. Math. Soc. 2001.

[8] MIGLIORE J.C., Introduction to liaison theory and deficiency modules, Progress in Mathematics 165,

Birkhauser, Boston 1998.

[9] MIGLIORE J.C. AND PETERSON C., A construction of codimension three arithmetically Gorenstein

subschemes of projective space, Trans. Amer. Math. Soc. 349 (9) (1997), 3803–3821.

[10] PESKINE C. AND SZPIRO L., Liaison des varietes algebrique. I, Invent. Math. 26 (1974), 271–302.

[11] RAGUSA A. AND ZAPPALA G., Properties of 3−codimensional Gorenstein schemes, Comm. Algebra29 (1) (2001), 303–318.

[12] RAGUSA A. AND ZAPPALA G., Gorenstein schemes on general hypersurfaces of Pr , Nagoya Math. J.

162 (2001), 111–125.

[13] ULRICH B., Sums of linked ideals, Trans. Amer. Math. Soc. 318 (1) (1990), 1–42.

AMS Subject Classification: 13H10.

Rosa Maria MIRO-ROIG, Departamento de Algebra y Geometria, Universidad de Barcelona, E-08007

Barcelona, ESPANA


Roberto NOTARI, Maria Luisa SPREAFICO, Dipartimento di Matematica, Politecnico di Torino, I-10129

Torino, ITALIA



Syzygy 2005

N. Mohan Kumar – A.P. Rao – G.V. Ravindra

FOUR-BY-FOUR PFAFFIANS


Abstract. This paper shows that the general hypersurface of degree ≥ 6 in projective four

space cannot support an indecomposable rank two vector bundle which is Arithmetically

Cohen-Macaulay and four generated. Equivalently, the defining polynomial of the hypersur-

face is not the Pfaffian of a four by four minimal skew-symmetric matrix.

1. Introduction

In this note, we study indecomposable rank two bundles E on a smooth hypersurface

X in P4 which are Arithmetically Cohen-Macaulay. The existence of such a bundle on

X is equivalent to X being the Pfaffian of a minimal skew-symmetric matrix of size

2k × 2k, with k ≥ 2. The general hypersurface of degree ≤ 5 in P4 is known to be

Pfaffian ([1], [2], [6]) and the general sextic in P4 is known to be not Pfaffian ([4]).

One should expect the result of [4] to extend to all general hypersurfaces of degree

≥ 6. (Indeed the analogous statement for hypersurfaces in P5 was established in [8],

see also [5].) However, in this note we offer a partial result towards that conclusion.

We show that the general hypersurface in P4 of degree ≥ 6 is not the Pfaffian of a

4 × 4 skew-symmetric matrix. For a hypersurface of degree r to be the Pfaffian of a

2k × 2k skew-symmetric matrix, we must have 2 ≤ k ≤ r . It is quite easy to show

by a dimension count that the general hypersurface of degree r ≥ 6 in P4 is not the

Pfaffian of a 2r × 2r skew-symmetric matrix of linear forms. Thus, this note addressesthe lower extreme of the range for k.

2. Reductions

Let X be a smooth hypersurface in P4 of degree r ≥ 2. A rank two vector bundle E

on X will be called Arithmetically Cohen-Macaulay (or ACM) if ⊕k∈ZHi (X, E(k))

equals 0 for i = 1, 2. Since Pic(X ) equals Z, with generator OX (1), the first Chernclass c1(E) can be treated as an integer t . The bundle E has a minimal resolution overP4 of the form

0→ L1φ→ L0 → E → 0,

where L0, L1 are sums of line bundles. By using the isomorphism of E and E∨(t), we

obtain (see [2]) that L1 ∼= L∨0 (t − r) and the matrix φ (of homogeneous polynomials)

can be chosen as skew-symmetric. In particular, L0 has even rank and the defining

polynomial of X is the Pfaffian of this matrix. The case where φ is two by two is justthe case where E is decomposable. The next case is where φ is a four by four minimal

471

472 N. Mohan Kumar – A.P. Rao – G.V. Ravindra

matrix. These correspond to ACM bundles E with four global sections (in possibly

different degrees) which generate it.

Our goal is to show that the generic hypersurface of degree r ≥ 6 in P4 does

not support an indecomposable rank two ACM bundle which is four generated, or

equivalently, that such a hypersurface does not have the Pfaffian of a four by four

minimal matrix as its defining polynomial.

So fix a degree r ≥ 6. Let us assume that E is a rank two ACM bundle which

is four generated and which has been normalized so that its first Chern class t equals 0

or −1. If L0 = ⊕4i=1OP(ai ) with a1 ≥ a2 ≥ a3 ≥ a4, the resolution for E is given by

4⊕i=1

OP(t − ai − r)φ

−→4⊕i=1

OP(ai ).

Write the matrix of φ as

φ =

0 A B C

−A 0 D E

−B −D 0 F

−C −E −F 0

.

Since X is smooth with equation AF − BE + CD = 0, the homogeneous entries

A, B,C, D, E, F are all non-zero and have no common zero on P4.

LEMMA 1. For fixed r and t (normalized), there are only finitely many possi-

bilities for (a1, a2, a3, a4).

Proof. Let a, b, c, d, e, f denote the degrees of the poynomials A, B,C, D, E, F .Since the Pfaffian of the matrix is AF − BE + CD, the degree of each matrix en-

try is bounded between 1 and r − 1. a = a1 + a2 + (r − t), b = a1 + a3 + (r − t) etc.Thus if i /= j , 0 < ai + a j + r − t < r while

∑ai = −r + 2t . From the inequality,

regardless of the sign of a1, the other three values a2, a3, a4 are < 0. But again using

the inequality, their pairwise sums are > −r + t , hence there are only finitely many

choices for them. Lastly, a1 depends on the remaining quantities.

It suffices therefore to fix r ≥ 6 , t = 0 or −1 and a four-tuple (a1, a2, a3, a4)and show that there is no ACM bundle on the general hypersurface of degree r which

has a resolution given by a matrix φ of the type (a1, a2, a3, a4), t .

From the inequalities on ai , we obtain the inequalities

0 < a ≤ b ≤ c, d ≤ e ≤ f < r.

We do no harm by rewriting the matrix φ with the letters C and D interchanged to

assume without loss of generality that c ≤ d.

PROPOSITION 1. Let X be a smooth hypersurface of degree ≥ 3 in P4 support-

ing an ACM bundle E of type (a1 ≥ a2 ≥ a3 ≥ a4), t . The degrees of the entries of φcan be arranged (without loss of generality) as:

a ≤ b ≤ c ≤ d ≤ e ≤ f.

Four-by-four Pfaffians 473

Then X will contain a curve Y which is the complete intersection of hypersurfaces

of the three lowest degrees in the arrangement and a curve Z which is the complete

intersection of hypersurfaces of the three highest degrees in the arrangement.

Proof. Consider the ideals (A, B,C) and (D, E, F). Since the equation of X is AF −BE + CD, these ideals give subschemes of X . Take for example (A, B,C). If thevariety Y it defines has a surface component, this gives a divisor on X . As Pic(X )= Z,there is a hypersurface S = 0 in P4 inducing this divisor. Now at a point in P4 where

S = D = E = F = 0, all six polynomials A, . . . , F vanish, making a multiple pointfor X . Hence, X being smooth, Y must be a curve on X . Thus (A, B,C) defines acomplete intersection curve on X .

To make our notations non-vacuous, we will assume that at least one smooth hy-

persurface exists of a fixed degree r ≥ 6 with an ACM bundle of type (a1 ≥ a2 ≥ a3 ≥a4), t . Let F(a,b,c);r denote the Hilbert flag scheme that parametrizes all inclusionsY ⊂ X ⊂ P4 where X is a hypersurface of degree r and Y is a complete intersec-

tion curve lying on X which is cut out by three hypersurfaces of degrees a, b, c. Ourdiscussion above produces points in F(a,b,c);r and F(d,e, f );r .

LetHr denote the Hilbert scheme of all hypersurfaces in P4 of degree r and let

Ha,b,c denote the Hilbert scheme of all curves in P4 with the same Hilbert polynomial

as the complete intersection of three hypersurfaces of degrees a, b and c. Following J.Kleppe ([7]), the Zariski tangent spaces of these three schemes are related as follows:

corresponding to the projections

F(a,b,c);rp2→ Ha,b,c

↓ p1

Hr

if T is the tangent space at the point Yi

↪→ X ⊂ P4 of F(a,b,c);r , there is a Cartesiandiagram

Tp2−→ H0(Y,NY/P)

↓ p1 ↓ α

H0(X,NX/P)β−→ H0(Y, i∗NX/P)

of vector spaces.

Hence p1 : T → H0(X,NX/P) is onto if and only if α : H0(Y,NY/P) →H0(Y, i∗NX/P) is onto. The map α is easy to describe. It is the map given as

H0(Y,OY (a) ⊕OY (b) ⊕OY (c))[F,−E,D]−−−−−−→ H0(Y,OY (r)).

Hence

PROPOSITION 2. Choose general forms A, B,C, D, E, F of degrees a, b, c,d, e, f and let Y be the curve defined by A = B = C = 0. If the map

H0(Y,OY (a) ⊕OY (b) ⊕OY (c))[F,−E,D]−−−−−−→ H0(Y,OY (r))


is not onto, then the general hypersurface of degree r does not support a rank two ACM

bundle of type (a1, a2, a3, a4), t .

Proof. Consider a general Pfaffian hypersurface X of equation AF − BE + CD = 0

where A, B,C, D, E, F are chosen generally. Such an X contains such a Y and X isin the image of p1. By our hypothesis, p1 : T → H0(X,NX/P) is not onto and (incharacteristic zero) it follows that p1 : F(a,b,c);r → Hr is not dominant. Since all

hypersurfaces X supporting such a rank two ACM bundle are in the image of p1, we

are done.

REMARK 1. Note that the last proposition can also be applied to the situation

where Y is replaced by the curve Z given by D = E = F = 0, with the map given by

[A,−B,C], with a similar statement.

3. Calculations

We are given general forms A, B,C, D, E, F of degrees a, b, c, d, e, f where a+ f =b + e = c + d = r and where without loss of generality, by interchanging C and D

we may assume that 1 ≤ a ≤ b ≤ c ≤ d ≤ e ≤ f < r . Assume that r ≥ 6. We will

show that if Y is the curve A = B = C = 0 or if Z is the curve D = E = F = 0, de-

pending on the conditions on a, b, c, d, e, f , either H0(NY/P)[F,−E,D]−−−−−−→ H0(OY (r))

or H0(NZ/P)[A,−B,C]−−−−−−→ H0(OZ (r)) is not onto. This will prove the desired result.

3.1. Case 1

b ≥ 3, c ≥ a + 1, 2a + b < r − 2.In P5 (or in 6 variables) consider the homogeneous complete intersection ideal

I = (Xa0 , Xb1, X

c2, X

r−c3 , Xr−b4 , Xr−a5 − Xc−a−12 Xr−c−a−13 Xa+24 )

in the polynomial ring S5 on X0, . . . , X5. Viewed as a module over S4 (the polynomialring on X0, . . . , X4), M = S5/I decomposes as a direct sum

M = N (0) ⊕ N (1)X5 ⊕ N (2)X25 ⊕ · · · ⊕ N (r − a − 1)Xr−a−15 ,

where the N (i) are graded S4 modules. Consider the multiplication map X5 : Mr−1 →Mr from the (r − 1)-st to the r -th graded pieces of M . We claim it is injective and notsurjective.

Indeed, any element m in the kernel is of the form nXr−a−15 where n is a

homogeneous element in N (r − a − 1) of degree a. Since X5 · m = n · Xr−a5 ≡n · Xc−a−12 Xr−c−a−13 Xa+24 ≡ 0 mod (Xa0 , X

b1, X

c2, X

r−c3 , Xr−b4 ) we may assume that

n itself is represented by a monomial in X0, . . . , X4 of degree a. Our inequalities havebeen chosen so that even in the case where n is represented by Xa4 , the exponents of X4in the product is a + a + 2 which is less than r − b. Thus n and hence the kernel must

be 0.


On the other hand, the element Xa−10 X21Xc−a−12 Xr−c−a−13 Xa+14 in Mr lies in its

first summand N (0)r . In order to be in the image of multiplication by X5, this elementmust be a multiple of Xc−a−12 Xr−c−a−13 Xa+24 . By inspecting the factor in X4, this is

clearly not the case. So the multiplication map is not surjective.

Hence dim Mr−1 < dim Mr . Now the Hilbert function of a complete intersec-

tion ideal like I depends only on the degrees of the generators. Hence, for any complete

intersection ideal I ′ in S5 with generators of the same degrees, for the correspondingmodule M ′ = S5/I

′, dim M ′r−1 < dim M ′

r .

Now coming back to our general six forms A, B,C, D, E, F in S4, of the samedegrees as the generators of the ideal I above. Since they include a regular sequence

on P4, we can lift these polynomials to forms A′, B ′,C ′, D′, E ′, F ′ in S5 which give acomplete intersection ideal I ′ in S5.

The module M = S4/(A, B,C, D, E, F) is the cokernel of the map

X5 : M ′(−1) → M ′.

By our argument above, we conclude that Mr /= 0.

Lastly, the map H0(OY (a) ⊕OY (b) ⊕OY (c))[F,−E,D]−−−−−−→ H0(OY (r)) has cok-

ernel precisely Mr which is not zero, and hence the map is not onto.

3.2. Case 2

b ≤ 2.

Since the forms are general, the curve Y given by A = B = C = 0 is a smooth

complete intersection curve, with ωY ∼= OY (a + b + c − 5). Since a + b ≤ 4, OY (c)is nonspecial.

1. Suppose OY (a) is nonspecial. Then all three of OY (a),OY (b),OY (c) are non-special. Hence h0(NY/P) = (a+b+c)δ +3(1−g) where δ = abc is the degree

of Y and g is the genus. Also h0(OY (r)) = rδ+1−g+h1(OY (r)) ≥ rδ+1−g.To show that h0(NY/P) < h0(OY (r)), it is enough to show that

(a + b + c)δ + 3(1− g) < rδ + 1− g.

Since 2g− 2 = (a+ b+ c− 5)δ, this inequality becomes 5δ < rδ which is trueas r ≥ 6.

2. SupposeOY (a) is special (so b+ c ≥ 5), butOY (b) is nonspecial. By Clifford’stheorem, h0(OY (a)) ≤ 1

2aδ + 1. In this case h0(NY/P) < h0(OY (r)) will be

true provided that

1

2aδ + 1+ (b + c)δ + 2(1− g) < rδ + (1− g)

or r > b+c2

+ 1δ + 5

2.

Since c ≤ r2and b ≤ 2, this is achieved if


r > 2+r/22

+ 1δ + 5

2which is the same as r > 14

3+ 4

3δ .

But c ≥ 3, so δ ≥ 3, hence the last inequality is true as r ≥ 6.

3. Suppose both OY (a) and OY (b) are special. Hence a + c ≥ 5. Using Clifford’s

theorem, in this case h0(NY/P) < h0(OY (r)) will be true provided that12(a + b)δ + 2+ cδ + (1− g) < rδ + (1− g).

This becomes r > 12(a + b) + 2

δ + c. Using c ≤ r2, a + b ≤ 4, and δ ≥ 3, this is

again true when r ≥ 6.

3.3. Case 3

c < a + 1.

In this case a = b = c and r ≥ 2a. Using the sequence

0→ IY (a) → OP(a) → OY (a) → 0,

we get h0(NY/P) = 3h0(OY (a)) = 3[(a+44

)− 3] while h0(OY (r)) ≥ h0(OY (2a)) =(

2a+44

)−3(a+44

)+3. Hence the inequality h0(NY/P) < h0(OY (r))will be true provided(

2a+44

)> 6

(a+44

)− 12. The reader may verify that it reduces to 10a4 + 20a3 − 70a2 −

200a + 7(4!) > 0 and the last inequality is true when a ≥ 3. Thus we have settled this

case when r ≥ 6 and a ≥ 3. If r ≥ 6 and a (and hence b) ≤ 2, we are back in the

previous case.

3.4. Case 4

2a + b ≥ r − 2 and r ≥ 82.

For this case, we will study the curve Z given by D = E = F = 0 (of degrees

r − c, r − b, r − a) and consider the inequality h0(NZ/P) < h0(OZ (r)).Since a, b, c ≤ r

2, 2a+2 ≥ r−b ≥ r

2, hence a ≥ r

4−1. Also b ≥ a and 2a+b ≥ r−2,

hence b ≥ r3− 2

3. Likewise, c ≥ r

3− 2

3.

Now h0(OZ (r − a)) = h0(OP(r − a)) − h0(IZ (r − a)) ≤(r−a+44

)− 1 etc.,

hence

h0(NZ/P) ≤(r−a+44

)+(r−b+44

)+(r−c+44

)− 3 ≤

( 3r4

+54

)+ 2

( 2r3

+ 143

4

)− 3

or h0(NZ/P) ≤ G(r), where G(r) is the last expression.

Looking at the Koszul resolution for OZ (r), since a + b + c ≤ 3r2

< 2r , the

last term in the resolution has no global sections. Hence h0(OZ (r)) ≥ h0(OP(r)) −[h0(OP(a)) + h0(OP(b)) + h0(OP(c))] ≥

(r+44

)−(a+44

)−(b+44

)−(c+44

)≥(r+44

)−

3( r2+44

), or h0(OZ (r)) ≥ F(r), where F(r) is the last expression.

The reader may verify that G(r) < F(r) for r ≥ 82.

3.5. Case 5

6 ≤ r ≤ 81, 2a + b ≥ r − 2, b ≥ 3, c ≥ a + 1.

We still have r4− 1 ≤ a ≤ r

2, r3− 2

3≤ b, c ≤ r

2. For the curve Y given by A = B =


C = 0, we can explicitly compute h0(OY (k)) for any k using the Koszul resolution forOY (k). Hence both terms in the inequality h0(NY/P) < h0(OY (r)) can be computedfor all allowable values of a, b, c, r using a computer program like Maple and the

inequality can be verified. We will leave it to the reader to verify this claim.

References

[1] ADLER A. AND RAMANAN S., Moduli of abelian varieties, Lecture Notes in Mathematics 1644,

Springer-Verlag, Berlin 1996.

[2] BEAUVILLE A., Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39–64.

[3] CHIANTINI L. AND MADONNA C., ACM bundles on a general quintic threefold, Le Matematiche 55

(2) (2000), 239–258.

[4] CHIANTINI L. AND MADONNA C., A splitting criterion for rank 2 bundles on a general sextic three-

fold, Internat. J. Math. 15 (4) (2004), 341–359.

[5] CHIANTINI L. AND MADONNA C., ACM bundles on general hypersurfaces in P5 of low degree,

Collect. Math. 56 (1) (2005), 85–96.

[6] ILIEV A. AND MARKUSHEVICH D., Quartic 3-fold: Pfaffians, vector bundles, and half-canonical

curves, Michigan Math. J. 47 (2) (2000), 385–394.

[7] KLEPPE J., The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Appli-

cations to curves in 3-space, Ph. D. Thesis, University of Oslo, Oslo 1981.

[8] MOHAN KUMAR N., RAO A.P. AND RAVINDRA G.V., Arithmetically Cohen-Macaulay bundles on

hypersurfaces, to appear in Commentarii Mathematici Helvetici.

AMS Subject Classification: 14F05

N. MOHAN KUMAR, G.V. RAVINDRA, Department of Mathematics, Washington University in St. Louis,

St. Louis, Missouri, 63130, USA


A.P. RAO, Department of Mathematics, University of Missouri-St. Louis, St. Louis, Missouri, 63121, USA,



Syzygy 2005

G. Restuccia

SYMMETRIC ALGEBRAS OF FINITELY GENERATED

GRADED MODULES AND s-SEQUENCES


Abstract. We study properties of the symmetric algebra of finitely generated graded modules

M on a Noetherian ring R, generated by s-sequences. For these modules we investigate the

Eisenbud-Goto conjecture. If R = K [X1, . . . , Xn ] is a polynomial ring over a field K and

M has linear syzygies, we consider the jacobian dual module of M in order to describe the

Rees algebra of M .

1. Introduction

The aim of this paper is to study an interesting class of finitely generated modules M

on a Noetherian ring R for which the initial ideal of the presentation ideal J of their

symmetric algebra is very simple. More precisely, with respect to a special order on

the variables that correspond to the generators of the module M , we have a good ex-

pression for the initial ideal of J . This area was investigated in [7], where the authors

computed some algebraic invariants of SymR(M) or their bounds in terms of specialideals of the ring R. The theory gives definitive results if R is the polynomial ring in

m variables on a field K of any characteristic by using the Grobner basis theory (in the

following K always denotes a field). Here we would like to study an application of

previous results essentially in two directions. We have many areas of applications and

this is only the starting point of investigation via s-sequences. The first is to test the

Eisenbud-Goto conjecture (EGC) for the symmetric algebra of a module M generated

by an s-sequence. After we have given formulations in this case, we begin to work in

this direction. For regular sequences of forms in the polynomial ring (which are strong

s-sequences) we prove the (EGC). If M has linear syzygies on the polynomial ring

R = K [X1, . . . , Xm], a nice construction of [12] leads to the jacobian dual module Nof M . N is a finitely generated module on the ring Q = K [Y1, . . . ,Yn] and we havethe isomorphism SymR(M) = SymQ(N ). Then it is interesting to ask, when M is gen-

erated by an s-sequence, if N is generated by an s-sequence and viceversa, and this is

the second area. In this context it is possible to describe the Rees algebra of the module

M as a quotient of the symmetric algebra by its torsion submodule. In particular, in

section 1 we give the definition of s-sequence introduced in [7], we recall some known

results and we formulate the Eisenbud-Goto conjecture for the symmetric algebra of a

finitely generated graded module M generated by an s-sequence on a Noetherian ring

in different ways. In particular, if R = K [X1, . . . , Xm] is the polynomial ring, we giveit in terms of the annihilator ideals of the s-sequence. As an application, we verify

(EGC) for a regular sequence of forms of R.

In section 2 we introduce the jacobian dual of a module M on a polynomial ring

479

480 G. Restuccia

R = K [X1, . . . , Xm]. This module can be defined if the presentation matrix of Mhas linear entries in the variables Xi and its interest appears in many fields of commu-

tative algebra. If this module N over the polynomial ring K [Y1, . . . ,Yn] is generatedby a strong s-sequence, we obtain that the torsion submodule of SymR(M) coincideswith the first annihilator ideal of the s-sequence generating N and it is an ideal of Q.

The Rees algebra of M , as a quotient of SymR(M) by its torsion submodule, can becomputed. As an example, we consider a monomial ideal with linear syzygies not gen-

erated by an s-sequence, whose jacobian dual is generated by an s-sequence. The idea

is to address our interest to many computations in this direction.

2. Preliminaries

Let R be any Noetherian ring and M a finitely generated R-module with generators

f1, . . . , fn . If we consider a presentation of M

Rmf→ Rn → M → 0,

then f is represented by an n×m matrix (ai j )with entries in R, 1 ≤ i ≤ n, 1 ≤ j ≤ m.

The symmetric algebra of M on R, SymR(M) = ⊕i≥0Si (M), where, for each i , Si (M)is the component of degree i of SymR(M), has a presentation:

0→ J → SymR(Rn) → SymR(M) → 0

and SymR(Rn) O R[Y1, . . . ,Yn] is the polynomial ring on R in the variables Y j , J isthe relation ideal of SymR(M), J = (g1, . . . , gm), with gi form of degree 1 in the Y j ,gi =

∑nj=1 ai j Y j , for i = 1, . . . ,m, then SymR(M) O R[Y1, . . . ,Yn]!J .

The main problem is how to compute standard algebraic invariants of the graded alge-

bra SymR(M) such as the dimension dim(SymR(M)), the multiplicity e(SymR(M)),the depth depth(SymR(M)) with respect to the graded maximal ideal SymR(M)+ =⊕i>0Si (M), the regularity reg(M), in terms of the corresponding invariants of specialquotients of the ring R.

The first three invariants are classical. For the last invariant, we recall that reg(M) isthe Castelnuovo-Mumford regularity of the graded module M . Its importance is briefly

indicated in Eisenbud-Goto theorem which is an interesting description of regularity in

terms of the graded Betti numbers of M ([2]).

They show that reg(M), when M is a graded finite R-module, where R =K [X1, . . . , Xm], measures the “complexity” of the minimal free resolution of M as

an S-module. Therefore regularity plays an important role in many fields of commuta-

tive algebra.

More precisely, if we consider a graded minimal free resolution of M on S

0→ F" → · · · → F1 → F0 → M → 0,

if bi is the maximum degree of the generators of the free module Fi , then

reg(M) = sup{bi − i, i ≥ 0}.

Symmetric algebras and s-sequences 481

In other words, reg(M) is the smallest integer m such that for every j , the j-th syzygymodule of M is generated in degree ≤ m + j (equivalently, reg(M) = sup{βi,i+ j /=0, for some i}, where βi," are the graded Betti numbers of M).If R is not a polynomial ring, the regularity of M can be infinite. A nice area of

investigation in commutative algebra is the study of graded homogeneous algebras A

generated in the same degree such that regA(A/m+) = 0 as an A-module and m+ isthe maximal graded ideal of A.

A computation of the previous invariants can be obtained for a finitely generated

R-module that is generated by an s-sequence f1, . . . , fn in the sense of [7]. Considerthe presentation of SymR(M)

SymR(M) = R[Y1, . . . ,Yn]/J.

The ideal SymR(M)+ is generated by the residue classes of the Yi that are called f ∗i ,because the variables Yi correspond to the generators of the module M = R f1 +· · · + R fn in the presentation of SymR(M). For every i = 1, . . . , n, we set Mi−1 =R f1 + · · · + R fi−1 and let Ii = Mi−1 :R fi be the colon ideal. We set I0 = (0) forconvenience. Since Mi/Mi−1 O R/Ii , so Ii is the annihilator of the cyclic moduleR/Ii , Ii is called an annihilator ideal of the sequence f1, . . . , fi .

Consider the polynomial ring R[Y1, . . . ,Yn] and let < be a monomial order on

the monomials of R[Y1, . . . ,Yn] in the variables Yi such that

Y1 < Y2 < · · · < Yn .

We call < an admissible order.

With respect to this term order, if f =∑aαY

α , Y α = Yα11 · · · Y αn

n , α ∈ Nn , we put

in< f = aαYα , where Y α is the largest monomial in f such that aα /= 0.

If we assign degree 1 to each variable Yi and degree 0 to the elements of R, we have

the following facts:

1) J is a graded ideal,

2) the natural epimorphism S → SymR(M) is a graded homomorphism of gradedalgebras on R, S is a graded ring and SymR(M) is a graded ring.

DEFINITION 1. The sequence f1, . . . , fn is an s-sequence for M if

(I1Y1, I2Y2, . . . , InYn) = in< J.

If I1 ⊆ I2 ⊆ · · · ⊆ In, the sequence is a strong s-sequence.

EXAMPLE 1. Any d-sequence of elements a1, . . . , an in R is a strong s-se-

quence, with respect to the reverse lexicographic order on the Yi , with Y1 < Y2 <· · · < Yn ([7], Cor. 3.3).

As a consequence regular sequences, proper sequences are strong s-sequences, since

they are d-sequences ([10]).

If I = (a1, . . . , an), we have SymR(I ) = RR(I ), the Rees algebra of I .

482 G. Restuccia

If R = K [X1, . . . , Xm] we can use the Grobner basis theory and Buchberger’salgorithm to compute in< J .

If R = K [X1, . . . , Xm], then SymR(M) = K [X1, . . . , Xm,Y1, . . . ,Yn]/J . We canintroduce a term order on

S = K [X1, . . . , Xm,Y1, . . . ,Yn]

such that Y1 < Y2 < · · · < Yn and Xi < Yi for any i .

For example X1 < X2 < · · · < Xm < Y1 < Y2 < · · · < Yn is such a term order.

If G is a Grobner basis for J ⊂ K [X1, . . . , Xm,Y1, . . . ,Yn], we have in< J

= (in<G) = (in< f, f ∈ J ) and if the elements of G are linear in the Yi s, it follows

that f1, . . . , fn is an s-sequence for M .

REMARK 1. If R = K [X1, . . . , Xm], from the theory of Grobner basis, if

f1, . . . , fn is an s-sequence with respect to any admissible term order<, then f1, . . . , fnis an s-sequence for another admissible term order, too.

THEOREM 1 ([7]). Suppose R is a standard graded algebra, M is a graded

R-module which is generated by the homogeneous s-sequence f1, . . . , fn, where all fihave the same degree, I1, . . . , In are the annihilator ideals of the sequence f1, . . . , fn.Then:

i) dimSymR(M) = max0≤r≤n

1≤r1≤···≤rn≤n

{dim R/(Ir1 + · · · + Irn ) + r}

ii) e(SymR(M)) =∑

0≤r≤n1≤r1≤···≤rn≤n

e(R/(Ir1 + · · · + Irn )),

where dim R/(Ir1 + · · · + Irn ) = d − r , d = dimSymR(M).

For a strong s-sequence we have:

d = dimSymR(M) = max0≤r≤n

{dim R/Ir + r},

e(SymR(M)) =∑

0≤r≤ndim R/Ir=d−r

e(R/Ir ).

THEOREM 2 ([7]). If R = K [X1, . . . , Xm] and M is generated by elements of

the same degree, which are a strong s-sequence, then

1) reg(SymR(M)) ≤ max{reg(R/Ii ), i = 1, . . . , n} + 1,

2) depth(SymR(M)) ≥ min{depth(R/Ii ) + i, i = 0, . . . , n}.

The notion of s-sequence can be useful essentially:

1) to test some conjectures for graded modules M generated by s-sequences,

“via” conjectures about annihilator ideals of M , in particular we are interested to test


the Eisenbud-Goto conjecture (EGC) for SymR(M), when M has generators of the

same degree and the regularity is the ordinary regularity.

The (EGC), that involves all invariants of SymR(M), can be more easily verifiedif M is generated by an s-sequence.

2) to describe the Rees algebra of the R-module M , R = K [X1, . . . , Xm],

RR(M) = SymR(M)/(SymR(M))0

and to test (EGC) in the case the matrix of the relations of M is linear in the variables

X1, . . . , Xm . In this situation in fact we have a nice construction that collects manycases of ideals and modules (in particular those ones with linear resolution): the jaco-

bian dual module N .

If N is the Jacobian dual of M , then natural questions arise:

i) When the jacobian dual N of M is generated by an s-sequence?

ii) If it is the case, does SymQ(N ) verify (EGC)?

3. Eisenbud-Goto conjecture

There are several conjectures to connect the measures of the complexity of an algebra.

One of the most important is the following:

CONJECTURE 2. (EGC) If A is a standard graded domain on a field K then

reg(A) ≤ e(A) − codim(A),

where codim(A) = emb dim(A) − dim(A).

If A is Cohen-Macaulay, the conjecture is true and we have equality ([2]).

We will establish the (EGC) for symmetric algebras of finitely generated graded mod-

ule M generated by s-sequences.

We consider different formulations of the conjecture.

1) (EGC1) Eisenbud-Goto conjecture for the symmetric algebra of a module M

on a standard graded algebra R, generated on R by a strong s-sequence of elements of

the same degree, and such that SymR(M) is a domain

reg(SymR(M)) ≤ e(SymR(M)) − codim(SymR(M))

2) (EGC2) If R = K [X1, . . . , Xm], M a graded R-module generated by a

strong s-sequence of elements of the same degree, Eisenbud-Goto conjecture for the

symmetric algebra of M in terms of the annihilator ideals of the strong s-sequence

generating M is

max{reg(R/Ii ) : i = 1, . . . , n} + 1 ≤n∑

i=1

e(R/Ii )− (n +m) + max0≤i≤n

{dim(R/Ii ) + i}

484 G. Restuccia

3) (EGC3) Eisenbud-Goto conjecture for any annihilator prime ideal of a strong

s-sequence of the same degree> 1, generating a graded R-module M , R = K [X1, . . . ,Xm], or R standard graded algebra that is a domain(EGCi ) reg(R/Ii ) ≤ e(R/Ii ) − codim(R/Ii ), for i = 1, . . . , n, dim R/Ii = d − i .

(EGC’i ) reg(R/Ii ) ≤ e(R/Ii ) − m + dim(R/Ii ), i = 1, . . . , n, dim R/Ii = d − i .

In 2) and 3) d = dimSymR(M).

4) The same conjecture formulated for the Rees algebraR(M), whenR(M) =SymR(M), R a standard graded domain and M generated on R by a strong s-sequence

of elements of the same degree, becomes:

(EGC1’) reg(R(M)) ≤ e(R(M)) − codim(R(M)).If R = K [X1, . . . , Xm], M a graded finitely generated R-module generated by an

s-sequence of elements of the same degree, Eisenbud-Goto conjecture of R(M) =SymR(M) in terms of annihilator ideals of the strong s-sequence generating M (for

example, M is an ideal of R = K [X1, . . . , Xm] generated by a d-sequence of elementsof R):

(EGC2’) max{reg(R/Ii ) : i = 1, . . . , n} ≤n∑

i=1

e(R/Ii ) − n

(EGC3’) reg(R/Ii ) ≤ e(R/Ii ) − m + dim(R/Ii )

Some implications:

(EGC2)⇒ (EGC1)

If R = K [X1, . . . , Xm], by Theorem 1 and Theorem 2, we have:

reg(SymR(M)) ≤ max{reg R/Ii : i = 1, . . . , n} + 1

≤n∑

i=1

e(R/Ii ) − (n + m) − max0≤i≤n

{dim(R/Ii + i)}

≤ e(SymR(M)) − codim(SymR(M))

(EGC’i )⇒ (EGCi ), for any ideal Ii , i = 1, . . . , n.

reg(R/Ii ) ≤ e(R/Ii ) − m + dim(R/Ii ) ≤ e(R/Ii ) − codim(R/Ii ).

(EGC’i ) for every i = 1, . . . , n ⇒ (EGC2)

Since (EGC’i ) is true for every i , i = 1, . . . , n, we have:

max{reg R/Ii : i = 1, . . . , n}

≤n∑

i=1

e(R/Ii ) − (n + m) − (n − 2)m + dim(R/Is) +n∑

i=1i /=s

dim(R/Ii )


≤n∑

i=1

e(R/Ii ) − (n + m) +n∑

i=1i /=s

(dim(R/Ii ) − m) + dim R/Is

≤n∑

i=1

e(R/Ii ) − (n + m) + dim R/Is + s − 1

since n ≤ m.

If s is the integer such that dim(R/Is) + s = max1≤i≤n

{dim(R/Ii ) + i}, then we

have

max{reg R/Ii : i = 1, . . . , n}

≤n∑

i=1

e(R/Ii ) − (n + m) − (n − 2)m + dim(R/Is) +n∑

i=1i /=s

dim(R/Ii )

≤n∑

i=1

e(R/Ii ) − (n + m) +n∑

i=1i /=s

(dim(R/Ii ) − m) + dim R/Is

≤n∑

i=1

e(R/Ii ) − (n + m) + dim R/Is + s − 1

since dim R/Ii − m ≤ 0, i = 1, . . . , n.

In order to state the (EGC) for the jacobian dual module N of M , we need some

facts on N . As a consequence we will give the formulation of (EGC) for N in the next

section.

EXAMPLE 2 (Regular sequences). Let R be a Noetherian ring and let f1, . . . , fnbe a regular sequence of elements of R. Then f1, . . . , fn is a strong s-sequence withrespect to any reverse lexicographic order on the variables Y1, . . . ,Yn such that Y1 <Y2 < . . . < Yn with annihilator ideals I1 = (0), I2 = ( f1), . . . , In = ( f1, . . . , fn−1)and in< J = (( f1)Y2, . . . , ( f1, . . . , fn−1)Yn).In fact a regular sequence is a d-sequence, hence the assertion follows.

THEOREM 3. Let R be a Noetherian ring and let f1, . . . , fn be a regular se-quence. Let I = ( f1, . . . , fn) and SymR(I ) = R[Y1, . . . ,Yn]/J . Then we have:

1) J is minimally generated by the elements gi j = fi Y j − f j Yi , 1 ≤ i < j ≤ n.

2) If R = K [X1, . . . , Xm], the set {gi j , 1 ≤ i < j ≤ n} is a Grobner basis withrespect to any reverse lexicographic order on the Y j and such that Y1 < . . . <Yn.

486 G. Restuccia

Proof. 1) Put gi j = fi Y j − f j Yi , i < j and suppose that {gi j }1≤i< j≤n is not a minimalsystem of generators of J . Then

fi Y j − f j Yi =∑

(ρ,k) /=(i, j)

hρkgρk,

fore some i, j , 1 ≤ i < j ≤ n.

Hence

fi =∑

ρ> j

h jρ fρ +∑

ρ< j

hρ j fρ

and this is a contradiction.

2)We have to consider the S-couples:

i) S(gi j , gik), j /= k

ii) S(gi j , gkj ), i /= k

iii) S(gi j , gkρ), i /= k, j /= ρ

For i), S(gi j , gik) = Yi (− fkY j + f j Yk).For ii), S(gi j , gkj ) = f j (− fi Yk + fkYi ).For iii), S(gi j , gkρ) = − fkYρgi j − f j Yi gkρ .

So, by Buchberger’s criterion we get in<(J ) = (in<gi j ).

Now, let R = K [X1, . . . , Xm] be a polynomial ring and let I be an ideal of Rgenerated by an R-sequence f1, . . . , fn of homogeneous elements.

Case I: f1, . . . , fn have the same degree a.

PROPOSITION 1. reg(SymR(I )) ≤ (n − 1)(a − 1) + 1.

Proof. Since f1, . . . , fn is a strong s-sequence, then we can apply the formula

reg(SymR(I )) ≤ max{reg(R/Ii ), 1 ≤ i ≤ n} + 1,

where I0 = I1 = (0), I2 = ( f1), . . . , In = ( f1, . . . , fn−1). The result follows by theKoszul resolution for the annihilator ideals Ii , 2 ≤ i ≤ n.

PROPOSITION 2. Let I = ( f1, . . . , fn) ⊂ R = K [X1, . . . , Xm] be generatedby a regular sequence of forms of the same degree a. Then (EGC2’) is true.

Proof. We have to prove that

max{reg(R/Ii ), 1 ≤ i ≤ n} ≤∑

1≤i≤n

e(R/Ii ) − n


that is

(n − 1)a − (n − 1) ≤n∑

i=1

ai−1 − n

(n − 1)a ≤ a + a2 + · · · + an−1.

The assertion follows.

PROPOSITION 3. Let I = ( f1, . . . , fn) ⊂ R = K [X1, . . . , Xm] be generatedby a regular sequence of forms of the same degree a ≥ 2. Then (EGC’i ) is true for i ,

1 ≤ i ≤ n, such that the annihilator ideal Ii is a prime ideal.

Proof. We have to prove that reg(R/Ii ) ≤ e(R/Ii ) − i + 1, i.e. (i − 1)a − (i − 1) ≤ai−1 − i + 1 and (EGC’i ) is true.

REMARK 2.

1) For n > 1, in Proposition 1 we have in fact equality. The result can follow from

the resolution of the algebra SymR(I ) = R(I ), by employing the Eagon-Northcottcomplex. Let S = K [X1, . . . , Xm; Y1, . . . ,Yn] and let F and G be finitely generated

free graded S-modules of rank 2 andm respectively. Consider a graded homomorphism

of degree zero g : G → F , g represented by the matrix(

f1 ... fnY1 ... Yn

).

We can write g : S(−a)n → S2 and we consider the Koszul complex arising from g.

K (g) : 0→n∧G ⊗ S(F)(−n) →

n−1∧G ⊗ S(F)(−n + 1) → · · ·

· · · → G ⊗ S(F)(−1) → S(F) → 0,

where S(F) = SymS(F) = S[T ] = S[T1, T2] and the differential

δ :i∧G ⊗ S(F)(−i) →

i−1∧G ⊗ S(F)(−i + 1)

is defined by

δ(t1 ∧ t2 ∧ . . . ∧ ti ⊗ f (T )) =i∑

j=1

(−1) j g(t j )t1 ∧ t2 ∧ . . . ∧ t j ∧ . . . ∧ ti ⊗ f (T ).

Since ht (J ) = n − 1, dimR(I ) = m + 1 = n + m − ht (J ) and J is perfect.The complex

D0(g) : 0→( 0∧

G ⊗ Sn−2(F))∗

→(G ⊗ Sn−3(F)(−1)

)∗ → · · ·

→( n−3∧

G ⊗ S1(F)(−n + 3))∗

→( n−2∧

G ⊗ S0(F)(−n + 2))∗

→ S → 0,

488 G. Restuccia

resolves S/J ([7], (2.16)).Since any generator of J has degree a+ 1, the shift in the place 1 is −a, that is a is theshift of the generators of the module (∧n−2G ⊗ S0(F))∗. Finally, the complex aboveis the dual of a Koszul complex. Hence:

reg(S/J ) = (n − 1)a − (n − 1) + 1.

2) If f1, . . . , fn is a regular sequence of n forms of degree a ≥ 2, SymR(I ) = R(I ),I = ( f1, . . . , fn) and reg(R) ≤ regR(I ) ≤ max{reg R + 1, reg R + n(a − 1)} ([5],Corollary 2.6).

For m = 1, reg R = 0, regR(I ) = 0. The assertion follows and (EGC) is true.

For m > 1, reg R = 0 and 0 ≤ regR(I ) ≤ max{1, n(a − 1)} = n(a − 1).

For n ≥ 3, what is needed is n(a − 1) ≤n∑

i=1

ai−1 − n + 1, na ≤n∑

i=1

ai−1 + 1. If we

write na = a + (a + a) + (n − 3)a, we have na ≤ a + a2 +∑n

i=4 ai−1 and (EGC) is

true.

Case II: f1, . . . , fn are forms of different degrees d1, . . . , dn , d1 ≤ d2 ≤ . . . ≤ dn .

Consider R as a graded ring by assigning to each variable Xi degree 0. Then S =R[Y1, . . . ,Yn] is a graded ring if we assign to each variable Yi degree 1. Let < be

a monomial order on the monomials in Y1, . . . ,Yn such that Y1 < Y2 < · · · <Yn . Since f1, . . . , fn is a regular sequence, it is a strong s-sequence and in< J =(I1Y1, . . . , InYn), Ii = ( f1, . . . , fi−1) for i = 1, . . . , n.As a consequence

regR(I ) = reg R[Y1, . . . ,Yn]/J ≤ reg R[Y1, . . . ,Yn]/in< J == reg R[Y1, . . . ,Yn]/(I1Y1, . . . , InYn) ≤ max{reg(R/Ii ), 1 ≤ i ≤ n} + 1.

But the last regularity is 0, since all matrices have entries of degree 0 in the mini-

mal graded resolution of any annihilator ideal Ii . We need R(I ) is a standard gradedalgebra for the formulation of (EGC).

4. Jacobian dual

Let R = K [X1, . . . , Xs] be a polynomial ring and let E be a finitely generated R-module with presentation :

Rmφ→ Rn → E → 0

where the entries of the n × m matrix A = (ai j ) that represents φ are homogeneouslinear forms.

The equations of the symmetric algebra of E , SymR(E) = S(E) are

f j =n∑

i=1

ai j Yi j = 1, . . . ,m.


There is a naive duality for S(E), obtained from rewriting the equations f j in the Xi ’svariables.

f j =n∑

i=1

ai j Yi =s∑

i=1

bi j Xi j = 1, . . . ,m

and B = (bi j ) is an s × m matrix of homogeneous linear forms in the Yi ’s variables.

We have:

At

Y1...Yn

= Bt

X1...Xs

=

f1...fm

.

Now we put Q = K [Y1, . . . ,Yn] and consider the cokernel N of the map

Qm 8→ Qs → N → 0,

where 8 is the map represented by B.

N defines the Jacobian dual module of E ([12], [14]).

EXAMPLE 3. We can write the relation f = (X1 − 2X2)Y1 + (X1 + X2)Y2 +X3Y3 as f = (Y1 + Y2)X1 + (−2Y1 + Y2)X2 + Y3X3.

REMARK 3. SymR(E) ∼= SymQ(N ).

EXAMPLE 4. Suppose that A ∼= B, in the sense that the two matrices A and B

have the same elements under the substitution Xi −→ Yi , n = s. Then R ∼= Q and

E ∼= N .

There is a nice situation that will be interesting in the following.

Let R = K [X1, . . . , Xn], I = m+ = (X1, . . . , Xn), SymR(m+) = R(m+) =K [X1, . . . , Xn; Y1, . . . ,Yn]/J , where J is generated by the binomials XiY j − X jYi ,

1 ≤ i < j ≤ n, the 2× 2-minors of the 2× n matrix

(X1 X2 ... XnY1 Y2 ... Yn

).

The binomials in the Xi ’s give the dual matrix B of the relation matrix A of m+under the substitution Xi → Y j , i, j = 1, . . . , n.Notice that the set of binomials is an universal Grobner basis for the ideal J and this

implies m+ is generated by an s-sequence linear in the Yi ’s and linear in the Xi ’s, too.

Another example is given by mi = (X1, . . . , Xi ), i < n.

SymR(mi ) = K [X1, . . . , Xn; Y1, . . . ,Yi ]/Ji

where Ji is generated by the binomials X"Ys − XsY", 1 ≤ " < s ≤ i , the 2× 2 minorsof the matrix (

X1 X2 ... XiY1 Y2 ... Yi

).

490 G. Restuccia

Put S = K [Y1, . . . ,Yi ], SymR(mi ) = SymS(N ) and X1, . . . , Xi is an s-sequence (itis a regular sequence) for mi and the sequence of 1-forms x

∗1 , . . . , x

∗i is an s-sequence

for the jacobian dual N of mi , where x1, . . . , xn are the residue classes of X1, . . . , Xnin S[X1, . . . , Xn]/Ji .

PROPOSITION 4. Let R = K [X1, . . . , Xm] be a polynomial ring and M a

graded R-module generated by forms f1, . . . , fn of the same degree. Suppose that therelation ideal J of SymR(M) is generated by forms that are linear in both sets of vari-ables X and Y , and let N be the jacobian dual of M generated by x1, . . . , xm, wherex∗1 , . . . , x

∗m are the images of the elements X1, . . . , Xm in the ring K [X1, . . . , Xm;

Y1, . . . ,Yn]/J .Suppose J has a Grobner basis linear in the X and Y variables with respect to the

reverse lexicographic order on all variables and to the two orders of variables Xm >. . . > X1 > Yn > . . . > Y1 and Yn > . . . > Y1 > Xm > . . . > X1.

Then M is generated by an s-sequence if and only if N is generated by an s-sequence.

Proof. It is a consequence of the previous facts.

REMARK 4. The strong case concerns J with a universal Grobner basis that is

linear in the X and Y variables with respect to any permutation of variables.

REMARK 5. If we know the Grobner basis of J that is linear in the variables X

and Y , with respect to the reverse lexicographic order and to the two orders of variables

Xm > . . . > X1 > Yn > . . . > Y1 and Yn > . . . > Y1 > Xm > . . . > X1, then we

can write the annihilator ideals of the sequences f1, . . . , fn and x1, . . . , xm by usinglemma 3.3 of [9].

The theorem gives the annihilator ideals for the s-sequence generating the jacobian

dual N of M , but the proof can be repeated to have the annihilator ideals of M .

EXAMPLE 5. Let I = (X2,Y 2, XY ) that is generated by an s-sequence ([7],Examples 1.5(1)). The jacobian dual N of I is generated by an s-sequence, too, but the

relation ideal J has a Grobner basis linear in the Xi ’s, but not linear in the Yi ’s ([7]).

Now consider SymR(M) = R[Y1, . . . ,Yn]/J ∼= Q[X1, . . . , Xm]/J =SymQ(N ). Let x∗1 , . . . , x

∗m be the images of X1, . . . , Xm mod J that we can consider

as the generators of N (we denote by x1, . . . , xm the generators of N ).

We recall some propositions:

PROPOSITION 5. Let I ⊂ R be an ideal generated by f1, . . . , fn. Then thefollowing conditions are equivalent:

1) f1, . . . , fn is a d-sequence;

2) (0 : f1) ∩ I = 0 and f2, . . . , fn is a d-sequence in R/( f1).


Proof. [7], Lemma 3.1.

PROPOSITION 6. Let M be an R-module generated by f1, . . . , fn. Then thefollowing conditions are equivalent:

1) f1, . . . , fn is a strong s-sequence with respect to the lexicographic order inducedby Yn > Yn−1 > . . . > Y1;

2) f ∗1 , . . . , f ∗n is a d-sequence in SymR(M).

Proof. [7], Theorem 3.2.

PROPOSITION 7. Let M be a finitely generated R-module, and let R be a do-

main. Then SymR(M) is a domain if and only if (SymR(M))0 = 0, where (SymR(M))0⊆ SymR(M) is the torsion submodule of M ([13]).

THEOREM 4. Suppose N is generated by a strong s-sequence x1, . . . , xm. Thenwe have

1. x∗1 , . . . , x∗m is a d-sequence in SymQ(N );

2. the ideal (0 : x∗1 ) is generated by elements of Q;

3. if (0 : x∗1 ) is a prime ideal then

SymQ(N )/(0 : x∗1 ) ∼= R(M).

Proof. 1) If N is generated by a strong s-sequence, then x∗1 , . . . , x∗m is a d-sequence in

SymR(N ) (by Proposition 6).Then (0 : x∗1 ) ∩ (x∗1 , . . . , x

∗m) = (0) and (0 : x∗1 ) is generated by polynomials in

Y1, . . . ,Yn and we have 2).3) Suppose (0 : x∗1 ) a prime ideal of Q = K [Y1, . . . ,Yn]. So SymQ(N )/(0 : x∗1 ) ∼=SymR(M)/(0 : x∗1 ) is a domain, then (0 : x∗1 ) ⊆ (SymR(M))0, where (SymR(M))0 isthe torsion submodule of SymR(M). Since R is a domain, (0 : x∗1 ) = (SymR(M))0 (byProposition 7), then SymR(M)/(0 : x∗1 ) ∼= R(M), the Rees algebra of M ([3]).

EXAMPLE 6.

M = I = (X21, X22, X1X2) ⊂ R = K [X1, X2], X2 > X1

R2ϕ→ R3→I → 0

A = (ai j ) =

X2 0

0 X1−X1 −X2

J = ( f1, f2), where

f1 = X2Y1 − X1Y3 = Y1X2 − Y3X1

492 G. Restuccia

f2 = X1Y2 − X2Y3 = Y3X2 − Y2X1

The sequence X21, X22, X1X2 is a strong s-sequence for the ideal I ([7], Ex. 1.5(1)).

Consider

B = (bi j ) =(

−Y3 Y2Y1 −Y3

).

If S = K [Y1,Y2,Y3], the jacobian dual module N of I is

0 −→ S2ψ

−→ S2 −→ N −→ 0

S( f1, f2) = Y3 f1 + Y1 f2 = (−Y 23 + Y1Y2)X1 = f3.

Then a Grobner basis w.r.t. X2 > X1 > Y3 > Y2 > Y1 is { f1, f2, f3},

in< J = ((Y1,Y3)X2, (Y23 − Y1Y2)X1).

I ∗0 = (0), I ∗1 = (Y 23 − Y1Y2), I∗2 = (Y1,Y3), and since I

∗1 ⊂ I ∗2 , x1, x2 is a strong

s-sequence for N .

From f3 = (−Y 23 + Y1Y2)X1, we have: (0 : x∗1 ) = (Y1Y2 − Y 23 ). In order to prove

(Y 23 − Y1Y2) is a prime ideal in SymQ(N ) we remark that J ′ = ( f1, f2,Y23 − Y1Y2)

is a prime ideal in SymQ(N ) if and only if (Y1Y2 − Y 23 ) is a prime ideal in SymQ(N ).But J ′ is the ideal generated by the 2× 2-minors of the generic matrix

(X1 Y3 Y1X2 Y2 Y3

).

Then the assertion follows and (Y1Y2 − Y 23 ) is a prime ideal in SymQ(N ) and

SymQ(N )/(0 : x∗1 ) ∼= SymR((X21, X22, X1X2))/(0 : x∗1 ) ∼= R(I ) ∼=

∼= R[Y1,Y2,Y3]/(−Y3X1 + Y1X2,−Y2X1 + Y3X2,Y1Y2 − Y32).

EXAMPLE 7 (Monomial square-free matroidal ideals). Now we follow the no-

tations used in [11, page 130].

Let I be a monomial ideal of K [x1, . . . , xn] with the minimal set of generators G(I ) ={x J1 , . . . , x Jt }, where x J = x

j11 · · · x jnn , J = ( j1, . . . , jn) and i = (0, 0, . . . , 1, . . . , 0).

We set |J | = j1 + · · · + jn .

We can associate the vector∑t

i=1 ai⊗x Ji to a syzygy of I∑t

i=1 ai xJi , where⊗means

⊗K .

For any monomial order on K [x1, . . . , xn], we will say that

xi ⊗ x J < xk ⊗ xk if xi xJ < xkx

k .

If u ∈ G(I ), we put νi (u) = ji , if xjii appears in the monomial u.

Now, let I be a monomial ideal for which all generators have the same degree. I is

matroidal if it satisfies the following exchange property ([24]):

For all u, v ∈ G(I ) and all i with νi (u) > νi (v), there exists an integer j with ν j (v) >ν j (u), such that x j (u/xi ) ∈ G(I ).


THEOREM 5. Let I be a matroidal square-free ideal with generators x J1 , . . . ,

x JN of the same degree. Then a minimal set of generators for the first syzygies of I has

the form

x j ⊗ x Ji − xt ⊗ x J" , j + Ji = t + J", x Ji , x J" ∈ G(I )

where j < t , t integer such that if Ji = (a1, . . . , an), ak = bk, k = t + 1, . . . , n andsuch that b j > a j , for some x

Jk ∈ G(I ), Jk = (b1, . . . , bn).

Proof. In the reverse lexicographic order we can suppose that x J1 > x J2 > · · · > x JN .

Let x Ji < x Jk . Then there exists an integer t such that am = bm for m = t + 1, . . . , n,Ji = (a1, . . . , an), Jk = (b1, . . . , bn) and at > bt . Hence there exists an integer j

with b j > a j such that u′ = x j (x

Ji /xt ) ∈ G(I ). Thus there is a syzygy of the formx j ⊗ x Ji − xt ⊗ x J" , x J" ∈ G(I ).

EXAMPLE 8.

I = (x1, x2)(x3, x4) = (x1x3, x1x4, x2x3, x2x4)

In the reverse lexicographic order and for x4 > x3 > x2 > x1

x4x2 > x3x2 > x4x1 > x3x1.

We consider the mapping Y1 → x J4 = x3x1, Y2 → x J3 = x4x1, Y3 → x J2 = x3x2,

Y4 → x J1 = x4x2.

The syzygies are:

x1 ⊗ x J2 − x2 ⊗ x J4 −→ f1 = x2Y1 − x1Y3;

x1 ⊗ x J1 − x2 ⊗ x J3 −→ f2 = x2Y2 − x1Y4;

x3 ⊗ x J3 − x4 ⊗ x J4 −→ f3 = x4Y1 − x3Y2;

x3 ⊗ x J1 − x4 ⊗ x J2 −→ f4 = x4Y3 − x3Y4.

The relations matrix of I is

x2 0 x4 0

0 x2 −x3 0

−x1 0 0 x40 −x1 0 −x3

and the dual matrix is

−Y3 −Y4 0 0

Y1 Y2 0 0

0 0 −Y2 −Y40 0 Y1 Y3

Consider the order x4 > · · · > x1 > Y4 > · · · > Y1, J has a Grobner basis lin-

ear in the xi variables. In fact J = ( f1, f2, f3, f4) and a Grobner basis of J is

494 G. Restuccia

G = { f1, f2, f3, f4, f5, f6} with f5 = x4(Y2Y3 − Y1Y4), f6 = x2(Y2Y3 − Y1Y4),in< J = ((Y3,Y4)x1, (Y3Y2)x2, (Y2,Y4)x3, (Y3Y2)x4), I

∗1 = (Y3,Y4), I

∗2 = (Y3Y2),

I ∗3 = (Y2,Y4), I∗4 = (Y3Y2).

The jacobian dual N of I is generated by an s-sequence x∗1 , . . . , x∗4 that is not a strong

s-sequence.

The torsion submodule of SymR(I ) can be read in the dual matrix:

0 : x∗1 =(∣∣∣∣

−Y1 Y2Y3 Y4

∣∣∣∣

)= (Y1Y4 − Y3Y2).

REMARK 6. The ideal I is not generated by an s-sequence. The Grobner basis

of J is not linear in the variables Y1, . . . ,Y4 for any admissible order such that Y4 >Y3 > Y2 > Y1. Then we are forced in this case to study the invariants of SymQ(N ) ∼=SymR(M) “via” the jacobian dual.The computation of the annihilator ideals of the s-sequence generating N can be done

by the lemma 3.2 of [9]. Moreover this lemma can be used to compute the annhilator

ideals of a generating s-sequence of M , changing the variables xi ’s with the Y j ’s and

for a term order on all variables xi ’s and Y j ’s, that is admissible for xi and for Y j , for

example xn > xn−1 > . . . > x1 > Yn > Yn−1 > . . . > Y1 or Yn > Yn−1 > . . . >Y1 > xn > xn−1 > . . . > x1.

In general, it is possible that M is not generated by an s-sequence and N is

generated by an s-sequence. If this is the case (Ex. 8), we can obtain the Rees algebra

of M by the quotient SymR(M)/I1∗, I1

∗ = 0 : x1∗ = (SymR(M))0, where I1∗ is a

prime ideal. Then (EGC) can be true forR(M) that is a domain,

(EGC1∗) reg(R(M)) ≤ e(R(M)) − codim(R(M)).

Moreover we have, if R = K [X1, . . . , Xm], M a graded finitely generated R-module,

N the jacobian dual on Q = K [Y1, . . . ,Yn], generated by an s-sequence of elements ofQ of the same degree, Eisenbud-Goto conjecture for the symmetric algebra SymQ(N ),in terms of the annihilator ideals I ∗1 , . . . , I ∗m of Q.(EGC2∗) max{reg(Q/I ∗i ) : i = 1, . . . ,m} + 1 ≤

∑mi=1 e(Q/I ∗i ) − (n + m)++max0≤i≤m{dim(Q/I ∗i ) + i}

(EGC∗i ) reg(Q/I ∗i ) ≤ e(Q/I ∗i ) − codim(Q/I ∗i ), for i = 1, . . . ,m.

(EGC∗i′) reg(Q/I ∗i ) ≤ e(Q/I ∗i ) − n + dim(Q/I ∗i ), for i = 1, . . . ,m.

EXAMPLE 9. I = (X2,Y 2, XY ), SymR(I ) verifies the inequality of (EGC2),I0 = I1 = (0), I2 = (X2), I3 = (X,Y ).

reg(R/I2) = 1, reg(R/I3) = 0, e(R/I2) = 2, e(R/I3) = 1

then:

max{reg(R/Ii ), i = 1, 2, 3} <

3∑

i=1

e(R/Ii ) − 2


is true. For the jacobian dual: I ∗1 = (Y 23 − Y1Y2), I∗2 = (Y1,Y3)

reg(Q/I ∗1 ) = 1, reg(Q/I ∗2 ) = 0, e(Q/I ∗1 ) = 2, e(Q/I ∗2 ) = 1

then the inequality:

max{reg(Q/I ∗i ), i = 1, 2} <

2∑

i=0

e(Q/I ∗i ) − 2+ 1

is verified.

REMARK 7. Let M be a graded module on R = K [X1, . . . , Xm], let N be thejacobian dual of M . Suppose that x1, . . . , xm is a strong s-sequence for N and (0 : x∗1 )is a prime ideal of Q. Then x∗1 , . . . , x

∗m is a d-sequence in R(M) = SymQ(N )/(0 :

x∗1 ).

Our attention actually applies to prove the conjecture via the annihilator ideals

of the jacobian dual of large classes of monomial ideals with linear resolution.

References

[1] BRUNS W. AND VETTER U., Determinantal rings, LNM 1327, Springer-Verlag, Berlin 1988.

[2] EISENBUD D. AND GOTO S., Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984),

89–133.

[3] EISENBUD D., HUNEKE C. AND ULRICH B., What is the Rees algebra of a module?, Proc. Amer.

Math. Soc. 131 (2003), 701–708.

[4] HERZOG J. AND HIBI T., Cohen-Macaulay polymatroidal ideals, preprint.

[5] HERZOG J., POPESCU D. AND TRUNG N.V., Regularity of Rees algebras, J. London Math. Soc. 65

(2) (2002), 320–338.

[6] HERZOG J., RESTUCCIA G. AND RINALDO G., Regularity and depth of the symmetric algebra,

Beitraege der Algebra und Geometrie, to appear (2004).

[7] HERZOG J., RESTUCCIA G. AND TANG Z., s-sequences and symmetric algebras, Manuscripta Math.

104 (2001), 479–501.

[8] HERZOG J., TRUNG N.V. AND ULRICH B., On the multiplicity of blow-up rings of ideals generated

by d-sequences, J. Pure Appl. Algebra 80 (1992), 273–297.

[9] HERZOG J., TANG Z. AND ZARZUELA S., Symmetric and Rees algebras of Koszul cycles and their

Grobner bases, Manuscripta Math. 112 (2003), 489–509.

[10] HUNEKE C., The theory of d-sequences and powers of ideals, Adv. in Math. 46 (1982), 249–279.

[11] ELIAS J., GIRAL J.M., MIRO-ROIG R.M. AND ZARZUELA S., Six Lectures on Commutative Alge-

bra, Progress in Mathematics 165, Birkhauser, 1998.

[12] SIMIS A., ULRICH B. AND VASCONCELOS W.V., Jacobian dual fibrations, Amer. J. Math. 115 (1)

(1993), 47–75.

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249.

[14] VASCONCELOS W.V., Arithmetic of Blowup Algebras. London Math. Soc. Lect. Note Series 195,

Cambridge University Press, Cambridge 1994.

496 G. Restuccia

AMS Subject Classification: 13A30, 13C15, 16W50.

Gaetana RESTUCCIA, Dipartimento di Matematica, Universita di Messina, Contrada Papardo, salita

Sperone 31, 98166 Messina, ITALIA


CONTENTS OF VOLUME 64 (2006)

Volume 64, N. 1

L. Poggiolini, On local state optimality of bang-bang extremal . . . . . . . . . 1

C. Prieur, Robust stabilization of nonlinear control systems by means of hybrid

feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

A. Ramos, New links and reductions between the Brockett nonholonomic inte-

grator and related systems . . . . . . . . . . . . . . . . . . . . . . . . . 39

L. Rifford, The stabilization problem on surfaces . . . . . . . . . . . . . . . . 55

R. C. Rodrigues – D. F. M. Torres, Generalized splines in Rn and optimal control 63

D. F. M. Torres – A. Yu. Plakhov, Optimal control of Newton-type problems of

minimal resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

E. Trelat, Singular trajectories and subanalyticity in optimal control and Hamilton-

Jacobi theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

M. Wyrwas, Analytic control systems and their properties related to observers . 111

Volume 64, N. 2

LEZIONI LAGRANGIANE

D. A. Bini, Numerical solution of large Markov chains . . . . . . . . . . . . . 121

PAPERS

M.G. Cimoroni – E. Santi, Some new convergence results and applications of a

class of interpolating-derivative splines . . . . . . . . . . . . . . . . . . 143

G. Harutjunjan, B.W. Schulze,Mixed problems and edge calculus symbol struc-

tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

T. Diagana, Integer powers of some unbounded linear operators on p-adic Hilbert

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

M. Dreher, The Kirchhoff Equation for the p–Laplacian . . . . . . . . . . . . . 217

Volume 64, N. 3

LEZIONI LAGRANGIANE

A. Milani, Dynamical systems: regularity and chaos . . . . . . . . . . . . . . . 239

INVITED LECTURES

J. Dhombres, Rhetorique et algebre au temps des lumieres. La question de

la nature des quantites imaginaires selon Euler, Daviet de Foncenex,

et Laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

PAPERS

M. Takagi, Equivalence between the maximal ideals of the extended Weyl Alge-

bras C[x, y, ξ, 1ξ ]〈∂x , ∂y〉 and those of the localized Weyl algebra

C[x, y]x 〈∂x , ∂y〉. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

C. Bouzar – R. Chaili, A Gevrey microlocal analysis of multi-anisotropic differ-

ential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

D. Fetcu, Harmonic maps between complex Sasakian manifolds . . . . . . . . 319

I. Dragomirescu, Approximate neutral surface of a convection problem for vari-

able gravity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Volume 64, N. 4

P. Salmon, The scientific work of Paolo Valabrega . . . . . . . . . . . . . . . . 343

SURVEY PAPERS

R. Achilles – M. Manaresi, Generalized Samuel multiplicities and applications 345

D.A. Buchsbaum, Alla ricerca delle risoluzioni perdute . . . . . . . . . . . . . 373

L. Chiantini, Vector bundles, reflexive sheaves and low codimensional varieties 381

E. Gover, Maximal Poincare series and bounds for Betti numbers . . . . . . . . 407

R. Hartshorne, Liaison with Cohen–Macaulay modules . . . . . . . . . . . . . 419

S. Nollet, Deformations of space curves: connectedness of Hilbert schemes . . 433

RESEARCH PAPERS

R.M. Miro-Roig – R. Notari – M.L. Spreafico, Properties of some Artinian

Gorenstein rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

N. Mohan Kumar – A.P. Rao – G.V. Ravindra, Four-by-four Pfaffians . . . . . . 471

G. Restuccia, Symmetric algebras of finitely generated graded modules and s-

sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

AUTHORS INDEX

Achilles R. . . . . . . . . . . . . . . . . . . . . . 343

Bini D.A. . . . . . . . . . . . . . . . . . . . . . . 121

Bouzar C. . . . . . . . . . . . . . . . . . . . . . . 305

Buchsbaum D.A. . . . . . . . . . . . . . . . . 373

Chaili R. . . . . . . . . . . . . . . . . . . . . . . . 305

Chiantini L. . . . . . . . . . . . . . . . . . . . . 381

Cimoroni M.G. . . . . . . . . . . . . . . . . . 143

Dhombres J. . . . . . . . . . . . . . . . . . . . . 273

Diagana T. . . . . . . . . . . . . . . . . . . . . . 199

Dragomirescu I. . . . . . . . . . . . . . . . . 331

Dreher M. . . . . . . . . . . . . . . . . . . . . . . 217

Fetcu D. . . . . . . . . . . . . . . . . . . . . . . . 319

Gover E. . . . . . . . . . . . . . . . . . . . . . . . 407

Hartshorne R. . . . . . . . . . . . . . . . . . . . 419

Harutjunjan G. . . . . . . . . . . . . . . . . . . 159

Manaresi M. . . . . . . . . . . . . . . . . . . . . 345

Milani A. . . . . . . . . . . . . . . . . . . . . . . 239

Mir’o R.M. . . . . . . . . . . . . . . . . . . . . . 451

Mohan Kumar N. . . . . . . . . . . . . . . . 471

Nollet S. . . . . . . . . . . . . . . . . . . . . . . . 433

Notari R. . . . . . . . . . . . . . . . . . . . . . . . 451

Plakhov A. Yu. . . . . . . . . . . . . . . . . . 79

Poggiolini L. . . . . . . . . . . . . . . . . . . . 1

Prieur C. . . . . . . . . . . . . . . . . . . . . . . . 25

Ramos A. . . . . . . . . . . . . . . . . . . . . . . 39

Rao A.P. . . . . . . . . . . . . . . . . . . . . . . . 471

Ravindra G.V. . . . . . . . . . . . . . . . . . . 471

Restuccia G. . . . . . . . . . . . . . . . . . . . . 479

Rifford L. . . . . . . . . . . . . . . . . . . . . . . 55

Rodrigues R.C. . . . . . . . . . . . . . . . . . 63

Salmon P. . . . . . . . . . . . . . . . . . . . . . . 343

Santi E. . . . . . . . . . . . . . . . . . . . . . . . . 143

Schulze B.W. . . . . . . . . . . . . . . . . . . . 159

Spreafico M.L. . . . . . . . . . . . . . . . . . . 451

Takagi M. . . . . . . . . . . . . . . . . . . . . . . 299

Torres D.F.M. . . . . . . . . . . . . . . . . . . 63, 79

Tr’elat E. . . . . . . . . . . . . . . . . . . . . . . . 97

Wyrwas M. . . . . . . . . . . . . . . . . . . . . . 111

Special issues

and Conference Proceedings

published in Rendiconti

Stochastic Problems in Mechanics (1982)

Linear Partial and Pseudo-differential Operators (1983)

Differential Geometry on Homogeneous Spaces (1983)

Special Functions: Theory and Computation (1985)

Algebraic Varieties of Small Dimension (1986)

Linear and Nonlinear Mathematical Control Theory (1987)

Logic and Computer Sciences: New Trends and Applications (1987)

Nonlinear Hyperbolic Equations in Applied Sciences (1988)

Partial Differential Equations and Geometry (1989)

Some Topics in Nonlinear PDE’s (1989)

Mathematical Theory of Dynamical Systems and ODE, I-II (1990)

Commutative Algebra and Algebraic Geometry, I-III (1990-1991)

Numerical Methods in Applied Science and Industry (1991)

Singularities in Curved Space-Times (1992)

Differential Geometry (1992)

Numerical Methods in Astrophysics and Cosmology (1993)

Partial Differential Equations, I-II (1993–1994)

Problems in Algebraic Learning, I-II (1994)

Number Theory, I-II (1995)

Geometrical Structures for Physical Theories, I-II (1996)

Jacobian Conjecture and Dynamical Systems (1997)

Control Theory and its Applications (1998)

Geometry, Continua and Microstructures, I-II (2000)

Partial Differential Operators (2000)

Liaison and Related Topics (2001)

Turin Fortnight Lectures on Nonlinear Analysis (2002)

Microlocal Analysis and Related Topics (2003)

Splines, Radial Basis Functions and Applications (2003)

Polynomial Interpolation and Projective Embeddings - Lecture Notes

of the School (2004)

Polynomial Interpolation and Projective Embeddings - Proceedings of

the Workshop of the School (2005)

Control Theory and Stabilization, I-II (2005-2006)

Syzygy 2005 (2006)

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