Koszul homology and blowing-up rings

J. Herzog1 Aron Simis2

W. Vasconcelos3

Contents

1 Introduction 2

2 Ideals of linear type 5

3 Ascent and descent 9

4 Approximation complexes 12

5 Depth and acyclicity 18

6 Sequences 21

6.1 Generalizations of regular sequences . . . . . . . . . . . . . . . . . . . . . . 21

6.2 Syzygetic ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.3 Coarse Cohen-Macaulayfication . . . . . . . . . . . . . . . . . . . . . . . . . 25

7 Depth of the Koszul homology modules 25

7.1 Ideals generated by proper sequences . . . . . . . . . . . . . . . . . . . . . . 25

7.2 Koszul homology and linkage . . . . . . . . . . . . . . . . . . . . . . . . . . 26

8 Dimension of symmetric algebras 29

9 Cohen–Macaulay blowing-up rings 32

10 Cohen–Macaulay symmetric algebras 34

11 The syzygetic reduction groups 39

12 Sequences and the approximation complexes 42

13 Linear resolutions 49

These notes are a combined and expanded account of the individual talks given bythe authors at the Trento conference. They aim at providing a framework for studyingarithmetical properties of Rees algebras, associated graded rings, symmetric algebras andthe theory of linear resolutions of ideals and modules.

0

1

2

3

1

We avail ourselves of the opportunity to make a fuller exposition by drawing liberally- and likely with considerable personal biases in view of the as yet unsettled state of thesubject - from various other sources.

For all of these sins, we are much grateful to the organizers of the conference, ProfessorsSilvio Greco and Giuseppe Valla.

1 Introduction

Given a Noetherian ring R and an ideal I, there are several algebras that conveniently codeinformation, of an analytic or algebraic nature, on the variety V (I) that I defines. Becausethese algebras are defined in terms of various conormal bundles it is not surprising thattheir properties mirror how V (I) sits inside Spec(R).

A brief catalog of these algebras, which will be loosely designated as ’blowing-up’ rings,is the following:

(a) R(I) = R[It] =⊕Ists =Ress algebra of I.

(b) R(I)∗ = R[It, t−1] = Rees∗ algebra of I.

(c) grI(R) =⊕Is/Is+1 = graded algebra associated to the I-adic filtration.

(d) Sym(I) = symetric algebra of I.

(e) Sym(I) = symetric algebra of the first conormal bundle I/I2.

If, besides, I is a prime ideal and I(s) denotes its s-th symbolic power - a procedure thatcan be adequately extend to any ideal - the following algebras may be added:

(f) Symb⊕I(s)ts = symbolic power algebra of I.

(g) Symb(I)∗ = Symb(I)[t−1] = symbolic∗ power algebra of I.

(h) gr′I(R) =⊕Is/Is+1 = graded algebra associated to the I-symbolic filtration.

Each of the listed algebras packs considerable information about the ideal I. Noteworthyare (a) for its role in the process of blowing-up a variety along a sub variety, and (f) in viewof Zariski’s main lemma on holomorphic functions. The last three algebras may fail to beNoetherian, thus making their usefulness less approachable.

There are several relationships among these algebras, expressed by algebra homomor-phisms. For instances, grI(R) = R(I)⊗R R/I = R(I)∗/(t−1), ou Sym(I/I2) = Sym(I)⊗RR/I, and several other less explicit ones.

One of our main themes will be the study of the comparisons involved in the canonicaldiagram:

Sym(I)α //

��

R(I) //

��

Symb(I)

��Sym(I/I2)

β // grI(R) // gr′I(R)

The approach to be used will be that of expressing the comparison in the first square bycertain differential graded algebras - labelled approximation complexes - that, among other

2

properties, reflect the obstructions for the morphisms α and β to be isomorphisms, Thesecomplexes are constructed over Sym(I) and Sym(I/I2), and are intimately connected to thehomology modules Hi(x;R) of the ordinary Koszul complex K(x), built from a sequence xgenerating the ideal I. This is a rather natural approach since Sym(I) and Sym(I/I2) aremore directly accessible than the other rings, but places considerable emphasis on retrievingproperties of the Koszul homology modules from the specifics of the presentation ideal.When successful this method allows for describing many arithmetical properties of theblowing-up rings R(I) and grI(R), and permits certain comparisons between R(I) andSymb(I) as well.

The main approximation complexes associated to the sequence x, M(x) and Z(x),display different sensitivities vis-a-vis K(x), being acyclic in situations much broader thanthe usual context of regular sequences. It has been found that certain generalizationsthereof, d-sequence and proper sequences, play here the role of ‘acyclic sequence‘. Thisturns out to be rather fruitful, for its connections to the theory of linear and algebraresolutions, multiplicities, coarse Cohen–Macaulayfication of local rings, and for spawningclasses of Cohen–Macaulay algebras arising from symmetric algebras.

Before we describe, in some detail, the contents of these notes, we point out that althoughvarious of the results could be stated in greater generality, the context will be that ofNoetherian rings. For notation, terminology, and basic references - especially when dealingwith Cohen–Macaulay rings - we shall use [25] and [29]. In addition, [29] along with [12]will be our sources for the more readily accessible properties of Koszul complexes.

An early focus of this study is the canonical homomorphism that exists between thesymmetric algebra of the ideal I, Sym(I), and the corresponding Rees ring R(I) :

α : Sym(I)→ R(I),

induced by identification Sym1(I) ' R(I)1 ' I. α is always a surjection, and the case whenit is also injective occurs often enough to deserve a designation. Thus we shall call I anideal of linear type if α is an isomorphism.

Initial examinations of this mapping were carried out by various authors ([33], [32],[39]), who pointed out several classes of ideals of linear type. In §2 we begin listing someof these ideals and introduce the first measure of comparison. That will simply be δ1(I) =ker (Sym(I) → I2). It is easy to express for certain determinantal ideals, and later it willbe interpreted cohomologically.

In §3 we discuss lifting theorems. These are of two kinds. The first revolves around asurprising discovery of Valla ([46]), to wit, if the reduction of

β : SymR/I(I/I2) = Sym(I/I2)→ grI(R)

is an isomorphism, then α is an isomorphism also.The other results compare the homomorphism α for the ideals I, I/(f) and (I, f). As an

application one obtains a result of Huneke ([21]) that Sym(I) = R(I) for ideals generated byd-sequences. This notion belongs to a family of extensions of the notion of regular sequencesand will play an ubiquitous role here.

Given a set of generators x = {x1, . . . , xn} of the ideal I and an R-module M, weconstruct in §4 the associated approximation complexes. These are derived from a doubleKoszul complex and are closely linked to K(x,M), the ordinary Koszul complex associated

3

to x and M. Two of then will be at the center of interest: the so-calledM and Z complexes.They are complexes of graded modules over the polynomial ring S = R[T1, . . . , Tn] :

M(x,M) : 0→ Hn ⊗ S[−n]→ . . .→ H1 ⊗ S[−1]→ H0 ⊗ S → 0

and

Z(x,M) : 0→ Zn ⊗ S[−n]→ . . .→ Z1 ⊗ S[−1]→ Z0 ⊗ S → 0

where Hi and Zi denote the homology and cycles of K(x,M). In contrast to K(x,M), whosehomology depends on the generating set x, the homology of either M or Z depends solelyon I.

When M = R, we have H0(Z) = Sym(I) and H0(M) = Sym(I/I2). There are sev-eral relationships between H(M) and H(Z) : It allows for instance to conclude that ifH1(M) = 0, then Sym(I) ' R(I). Thus these complexes serve as an obstruction theory forthe homomorphism α.

In §5 we consider conditions for the acyclicity of these complexes (see also section 12).Those depend on having sufficiently high depth on the Koszul homology modules Hi. It willprepare the ground for one of our main applications in section 10. Here, as illustration, weconsider instances of ideals that are nearly enough to complete intersections.

Various extensions of the notion of regular sequence, e.g. relative regular sequence, d-sequence and proper sequence, are take up in §6. They exist in more profusion than regularsequences. For instance, we prove that any ideal of height n contains a d-sequence of heightn. This will be used to provide a coarse form of Cohen-Macaulayfication of local rings. Therelationships among these sequences will be looked at and new measures of syzygeticityintroduced.

In §7 we discuss methods for determining the depth of the Koszul homology modulesof an ideal. The main result here is a theorem of Huneke ([24]), stating how that depthmay change along the linkage class of the ideal. Around this result one can organize severalearlier ones of ideals with Cohen–Macaulay Koszul homology.

The next section develops a tool for reading the Krull dimension of a module M formthe details of a free presentation. It works often enough for the uses in §9 and §10.

In §10 we present some of the main applications of the approximation complexes so far.It is proved that, if R is Cohen–Macaulay and the Koszul homology of the ideal I is Cohen–Macaulay and a natural local bound on the number of generators hold, then Sym(I) ' R(I)and Sym(I/I2) ' grI(R). Furthermore, all of these algebras are Cohen–Macaulay, and if Ris Gorenstein then then grI(R) Gorenstein also.

By weakening some of the hypotheses of §9 - that reflect requirements for the acyclicityof the M - one is able in §10 and using mostly the Z-complex to obtain new classes ofCohen–Macaulay symmetric algebras and develop a theory of their specializations.

In §11 we introduce a new array of syzygetic groups associated to a sequence x and anR-module M. They extend the δ′ s of §6 and reflect the properties of the Koszul homologynot of the ideal I = (x) alone but of the irrelevant ideals of the graded algebras R(I) andgrI(R). It is used to provide the precise conditions for an ideal to be of linear type ([27])and to prepare the ground for §12. This section contains proofs of the following structuralaspects of the approximation complexes. Essentially (i.e. locally and modulo a faithfullyflat extension): (a) The complex M(I;M) is acyclic if and only if I is generated by a

4

d-sequence relative to M ; (b) (cf. [27]) The complex Z(I;M) is acyclic if and only if I isgenerated by a proper sequence relative to M.

Finally, in §13, we study theM-complex relative to the maximal ideal of a regular localring and its connections to the theory of linear resolutions. It allows for finding algebraresolutions of certain ideals, determination of Hilbert-Poincare series of resolutions anddisplay the lack of rigidity of the approximation complexes themselves.

2 Ideals of linear type

In this section we begin to introduce measures of comparison between the symmetric andRees algebra of ideals. Let R be a commutative ring and let I be an ideal of R. Denote byα the canonical surjection of R-algebras

0→ A = ker (α)→ Sym(I)α→ R(I)→ 0

If a = {a1, . . . , an} is a generating set for I, the morphism α can also be described in thefollowing manner. Present I as

0→ Z1 → Rnϕ→ I → 0.

ϕ induces a surjection of the polynomial ring R[T1, . . . , Tn] onto Sym(I), with Ti mappinginto ai ∈ Sym1(I). Its kernel is the ideal Q generated by the elements in Z1, that is, Qis generated by all linear forms

∑ciTi such that

∑ciai = 0. R(I) may also be viewed

as a homomorphic image of R[T1, . . . , Tn], with Ti mapping into ait. The kernel, Q∞, isgenerated by all forms F (T1, . . . , Tn) such that F (a1, . . . , an) = 0. We thus have a diagram

Q∞

&&

Q

xxR[T1, . . . , Tn]

%%xxSym(I) // R(I)

with A = ker (α) = Q∞/Q. It is clear, from this discussion, that alternative ways to presentthis comparison are needed. This difficulty will be somewhat dealt with in the next sections,especially with the introduction of the approximation complexes.

The case when α is an isomorphism occurs often enough to deserve its own designation.The following terminology was suggested by Robbiano and Valla.

Definition 2.1. The ideal I is said to be of linear type if the corresponding morphism αis an isomorphism.

The module–theoretic properties of Sym(I) play a dominant role in the study of thecomparison morphism α, as witnesses the following

Proposition 2.2. Let I be an ideal such that I ∩ (0 : I) = (0). The following conditionsare equivalent:

(a) I is of linear type.

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(b) Sym(I) has no I-torsion (i.e., 0 is the only element of Sym(I) annihilated by I.)

Proof. (a)⇒(b) is clear since, by hypotheses, R(I) has no I-torsion. Conversely, supposew ∈ Symt(I); since

Itw = α(w)Symt(I),

we easily get the desired assertion.Note that, as a consequence, if I ∩ (0 : I) = (0) and Sym(I) ' R(I) under some other

R-morphism, then α is an isomorphism as well.The first large class of ideal of linear type is provided by the following construction of

Salmon ([39]; see also [18]).

Example 2.3. Let E be a module over the ring A and let R = SymA(E). Pick for I theaugmentation ideal of R,

I = R+ =⊕t>0

Symt(E).

We have the natural R-morphism

E ⊗A SymA(E)→ SymA(E)+ → 0,

whose kernel is generated by the elements of the form

e⊗ f · w − f ⊗ e · w

, e, f ∈ E = Sym1(E), w ∈ SymA(E). Taking the n-th symmetric power of this homomor-phism - and using universal property of such powers with respect to the change of ringsA→ R - we get

Symn(E ⊗A R) = Symn(E)⊗A R→ Symn(I)→ 0

from which we obtain In ' Symn(I) when the relations above are taken into account.

An important module–theoretic obstruction to Sym(I) ' R(I) is provided by

Proposition 2.4. Let I be an ideal of the Noetherian ring R. If Sym(I) ' R(I), then foreach prime ideal P ⊃ I, IP can generated by ht(P ) elements.

Proof. We may assume that (R,P ) is a local ring of dimension d. We work with the derivedhomomorphism Sym(I/I2) ' grI(R).

Let R(I)∗ be the “large” Rees ring, that is R(I)∗ = R[It, u], u = t−1; then grI(R) =R(I)∗/(u). Since dimR(I)∗ ≤ d+1 ([?]) and u is a regular element, we have dim Sym(I/I2) ≤d. In this case its homomorphic image SymR/P (I/PI) has dimension at most d, as desired.

To compare, in general, the algebras Sym(I) and R(I), it is often useful to look at thegraded components of the morphism α. Thus, denote

αt : Symt(I) = St(I)→ It = R(I)t.

Since α0 and α1 are the ordinary identifications, the first place these algebras may differ isfor t = 2.

6

In order to spell out A2 = kerα2 we bring in the Koszul complex on a set of generatorsof the ideal I. This idea will play an ever increasing role as we go on. We shall refer to [29]and [12] for basic properties of the Koszul complexes.

Tensoring by R/I the implied presentation of I yields an exact sequence of R/I-modules

0→ Z1 ∩ IRn/IZ1 → Z1/IZ1 → Rn/IRn → I/I2 → 0.

Thus Z1/Z1 ∩ IRn is the first-kind syzygy module of the conormal module I/I2. It fits intothe exact sequence

0→ Z1 ∩ IRn/B1 → H1 → Z1/Z1 ∩ IRn → 0

where H1 (resp. B1) is the first homology module (resp. 1-boundaries) of the Koszulcomplex on the chosen set of generators of I. We set H∗1 = Z1/Z1 ∩ IRn and call it thesyzygy part of H1 (cf. [42]; see also §4).

Although H1 and H∗1 both depend on the set of generators for I, Z1 ∩ IRn/B1 dependsonly on I. To put this in evidence, we denote Z1 ∩ IRn/B1 by δ1(I).

Proposition 2.5. There exists a natural homomorphism of R/I-modules σ :∧2 I →

TorR1 (I,R/I) (”antisymmetrization”) such that δ1(I) = TorR1 (I,R/I)/σ(∧2 I).

Proof. Since I annihilates H1, IZ1 ⊂ B1. We thus see that δ1(I) fits naturally into theexact sequence

0→ B1/IZ1 → Z1 ∩ IRn/IZ1 → δ1(I)→ 0.

By the long exact sequence of Tor, as applied to the initial presentation of I, we haveTorR1 (I,R/I) = Z1 ∩ IRn/IZ1. But TorR1 (I,R/I) can also be viewed as the kernel of themultiplication mapping I ⊗ I → I2, a fact we shall now use. Let then σ :

∧2 I → I ⊗ Ibe the antisymmetrization mapping, that is, σ(u ∧ v) = u ⊗ v − v ⊗ u, for u, v ∈ I.The composite

∧2 I → I ⊗ I → I2 is zero, so that we actually have a homomorphismσ :∧2 I → TorR1(I,R/I).

Let I = (a1, . . . , an) and let {e1, . . . , en} be a basis of Rn such that ϕ(e1) = ai. Thehomomorphism

2∧I → TorR1(I,R/I) = Z1 ∩ IRn/IZ1

as applied to the element ai ∧ aj gives the element ajei − aiej( mod IZ1), with belongs toB1/IZ1. The composite is evidently onto, so σ(

∧2 I) = B1/IZ1.

The claim then follows from the exact sequence above.

Remark 2.6. Since the image of σ is killed by I, σ factors through∧2 I⊗R/I =

∧2(I/I2).On the other hand, if 2 is invertible, then σ is injective and gives, actually, the splittingTorR1(I,R/I) = δ1(I)⊕

∧2(I/I2).

Corollary 2.7. δ1(I) = A2 = ker (S2(I)→ I2).

Proof. We show that ker (S2(I)→ I2) = TorR1(I,R/I)/σ(∧2 I). To see this note that the

sequence2∧I → I ⊗ I → S2(I)→ 0

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is exact, where the last moping is the canonical one. But σ(∧2 I) ⊂ TorR1(I,R/I), and

therefore we get the commutative diagram

0 // TorR1(I,R/I) // I ⊗ I // I2 // 0

∧2 I //

OO

I ⊗ I

OO

// S2(I)

OO

// 0

from with the assertion follows.

To consider the relative strength of the vanishing of δ1(I) we call on two examples.

Proposition 2.8. (cf. [43]) Let (R,m) be a local ring of depth two and let I be an ideal ofheight two and projective dimension one. The following are equivalent:

(a) δ1(I) = 0.

(b) I is generated by a regular sequence of two elements.

Proof. Let

0→ F1 → F0 → I → 0

be a minimal projective resolution of I. Assume δ1(I) = 0; we must show that F1 has rankone. Associated to this resolution there is a canonical complex [50]:

0→2∧F1 → F1 ⊗ F0 → S2(F0)→ S2(I)→ 0.

From the minimality hypotheses and the construction of this complex, it is a minimalprojective resolution of S2(I). Since δ1(I) is the torsion-submodule of S2(I) (cf. (2.7)), andthe depth two condition, it follows that δ1(I) = 0 implies

∧2 F1 = 0, as desired.

The converse is obvious.

Remark 2.9. The proof also shows the following. Let R be a Noetherian ring and let Ibe an ideal of height two and projective dimension one. Then δ1(I) = 0 if and only if Iis generically a complete intersection. In particular, making use Macaulay’ s examples, weconclude that if R = k[x, y, z], there is no bound for the minimal number of generators ofideals with δ1(I) = 0

Proposition 2.10. Let R be a regular local ring and let I be a Gorenstein ideal of height3. If 1/2 ∈ R, then δ1(I) = 0.

Proof. It was proved in [13] (without the hypothesis 1/2 ∈ R) that for such ideals I/I2

is a Cohen–Macaulay module and that if x is a system of parameters for S = R/I, thenλ(I/I2 ⊗ S/(x)) = 3λ(S′), where S′ = S/(x) and λ = length. On the other hand, it followsfrom [?] that

∧2 I has projective dimension 3. Since 1/2 ∈ R,∧2 I =

∧2(I/I2) and thus∧2(I/I2) is a Cohen–Macaulay module. Furthermore, since it is a specialization of thegeneric case, it follows from the methods of [13] that

λ(2∧

(I/I2)⊗ S′) = 3∧

(S′).

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Now let0→ R→ Rn → Rn → I → 0

be a minimal resolution of I. tonsuring with S and taken into account that the entries ofthe las ’matrix’ lie in I, we get the exact sequence

0→ TorR1(I, S)→ Sn → Sn → I/I2 → 0.

Since I/I2 is Cohen–Macaulay, TorR1(I, S) will also be Cohen–Macaulay. Now to showδ1(I) = 0 note that we have the decomposition

TorR1(I, S) =2∧

(I/I2)⊕ δ1(I).

Tensoring its with S′ and comparing lengths, we get δ1(I)⊗ S′ = 0, and thus δ1(I) = 0 byNakayama lemma.

Later a whole array of measures of comparison will be introduced and convenientlyinterpreted cohomologically.

3 Ascent and descent

The comparison of Sym(I) and R(I) is enhanced by the existence of certain natural map-pings defined on Sym(I). One of these - later to the explained as a homology connectinghomomorphism - is the dowgrading homomorphism λ,

λt : St+1(I)→ St(I)

that sends b1 · b2 · · · bt+1 ∈ St+1 into the element b1(b2 · · · bt+1) ∈ St(I). It is clear that itdoes not matter with element is pulled out of the pack. The image of λt is the submoduleI(St(I)) of St(I), that is, the kernel of the canonical homomorphism St(I)→ St(I/I

2).We thus have the commutative exact diagram

0

��

0

��

0

��At+1

//

��

At //

��

At/λ(At+1) //

��

0

St+1(I)λt //

αt+1

��

St(I) //

αt

��

St(I/I2) //

βt��

0

0 // It+1 //

��

It //

��

It/It+1 //

��

0

0 0 0

The following surprising lifting property was discovered by Valla ([46]; see also [18])

Theorem 3.1. Let I be an ideal of the Noetherian ring R. Then α : Sym(I)→ R(I) is anisomorphism if and only if the reduction β : Sym(I/I2)→ grI(R) is an isomorphism.

9

Proof. We only have to show that β lifts to an isomorphism. The preceding diagram andthe snake lemma leads to the exact sequence:

0 // At/λ(At+1) // St(I/I2) // It/It+1 // 0 .

By hypotheses we have At = λ(At+1) for t ≥ 2. Since A is a finitely generated homogeneousideal of Sym(I), there exists an integer s ≥ 2 such that

At+1 = S1(I) · At, t ≥ s.

Applying λ to this equality and taking into account the hypothesis above, we get

At = λ(At+1) = λ(S1(I) · At) = I(At)

and thus, by localization and Nakayama lemma, At = 0 for t ≥ 0.By an obvious descending induction we get At = 0 for all t.Similar is the following (cf. [18]).

Theorem 3.2. A = ker (α) is nilpotent if and only if A∗ = ker (β) is nilpotent.

Proof. Since A is finitely generated, the equation of (2.2) implies that there exists aninteger r such that IrA = 0.

We only have to prove that A is nilpotent along with A∗. For this we again refer tothe earlier diagram. As A∗ = ker (β) is finitely generated, there exists a integer s suchthat [At/λ(At+1)]

s = 0, that is, Ast ⊂ λ(Ast+1) ⊂ IAst. Using the integer r we now getArst ⊂ IrArst = 0, as desired.

The next result (cf. [46]) is useful in showing that certain ideals generated by specialsequences are of linear type.

Proposition 3.3. Let I be an ideal of the ring R. Let f be an element of I such that foreach integer t, It∩(0 : f) = (0 : f)It. Se SymR/(f)(I/(f)) ' R(I/(f)), then Sym(I) ' R(I).

Remark 3.4. The condition is satisfied if I ∩ (0 : f) = 0, in particular if f is a regularelement.

Proof. From the inclusion (f) ⊂ I consider the exact sequence of symmetric algebras withits natural mappings into the corresponding Rees rings:

0 // (f) · Sym(I) //

��

Sym(I)+ //

α

��

Sym(I/(f))+ //

��

0

0 // (ft)R(I) // R(I)+ // R(I/(f))+ // 0

where all the vertical mappings are induced induced by α. Since we are assuming the iso-morphism on the right, the bottom row is exact as well. To show that α is an isomorphism,assume 0 6= w is a homogeneous element of least degree in A = ker (α). From the ex-actness of the top row, we have w = f · w′, w′ ∈ St−1(I). Since α(w) = 0 = fα(w′),α(w′) ∈ It−1 ∩ (0 : f) = (0 : f)It−1, and thus w′ = u + v, with u ∈ (0 : f)St−1 andv ∈ ker (αt−1). In this case, w = f · v = 0 by the minimality of t.

For application, we recall the notion of a d-sequence. It is a broad generalization of‘regular sequence’ and has been developed into a theory by Huneke ([20]). Along withseveral other notions of sequence it will be given a close scrutiny in later sections.

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Definition 3.5. A sequence x = {x1, . . . , xn} of elements in the ring R is called a d-sequence if:

(a1) x is a minimal generating set of elements for the ideal I = (x).

(a2) (x1, . . . , xi) : xi+1xk = (x1, . . . , xi) : xk, para i = 0, . . . , n− 1, k ≥ i+ 1.

Note that regular sequences are prime examples of d-sequences. For n = 1, {x} is ad-sequence if and only if 0 : x = o : x2, that is (0 : x) ∩ (x) = 0.

Theorem 3.6. (cf. [21]; see also [46]) Let I be an ideal generated by a d-sequence. ThenSym(I) ' R(I).

To prove this, via 3.3 and induction, we only need.

Proposition 3.7. (cf. [20]) If x = {x1, . . . , xn} is a d-sequence, then

(a) The images of x2, . . . , xn in R/(x1) form a d-sequence.

(b) (0 : x1) ∩ (x1, . . . , xn) = 0

Proof. (a) is immediate from the definition. We prove (b) by induction on n. By theremark above we assume n > 1. Let

∑rixi ∈ (0 : x1). As (0 : x1) ⊂ (0 : x1xn) = (0 : xn),

we have rnx2n ∈ (x1, . . . , xn−1) and hence rnxn ∈ (x1, . . . , xn−1) as required.

Remark 3.8. There exist ideals of linear type which are not generated by a d-sequence (cf.Example 6.7).

Corollary 3.9. (cf. [32]) Let I be an ideal generated by a regular sequence. Then Sym(I) 'R(I).

Corollary 3.10. (cf. [32]) Let (R,m) be a Noetherian local ring. Then Sym(I) ' R(I) ifand only if R is regular.

Proof. If R is regular, m is generated by a regular sequence and thus it is of linear typeby 3.9.

For the converse, note that by 2.5, δ1(m) = 0. This means that every element in Z1

with coefficients in m is a boundary. But if we consider a minimal presentation of m in 2.5,Z1 ⊂ mRn, and therefore H1 = 0. Thus, by [29], m is generated by regular sequence.

Proposition 3.11. (cf. [46]) Le J = (I, f) be an ideal of linear type. If f lies in theJacobson radical of R and I : f = I, then I is also of linear type.

Proof. Consider the presentation

0→ L→ I ⊕R ϕ→ J → 0

where ϕ(a, r) = a+ rf. Since I : f = I, L = {(−bf, b); b ∈ I}.The associated diagram of symmetric algebras and corresponding Rees rings is the fol-

lowing:

11

0 // G //

��

Sym(I)[U ] //

α[U ]

��

Sym(J) //

��

0

0 // H // R(I)[U ] // R(J) // 0

where G generated by the elements bU − fb ∈ Sym(I)[U ], b ∈ I, b = corresponding 1-formin S1(I), and H by the polynomials of bi-degree 1 bU − fbT ∈ R(I)[U ].

As in 3.3, from the isomorphism at the right we have the exactness of the bottom row.Let w be a homogeneous element in At = ker (αt). Then w ∈ G and thus

w =

n∑i=1

(biU − fbi)(t−1∑t=0

cijUj), cij ∈ St−1−j(I).

As w ∈ St(I), we must have the equations:

w =∑bici0

0 =∑bici0 − fbici1

...0 =

∑bici,t−1

They show, successively, that∑bici,j lies in A, and consequently At ⊂ fAt.

Remark 3.12. A converse of this result is proved in [46], under the condition It : f = It

for all t.

Another aspect of ideals generated by d-sequences that they satisfy the Artin–Reeslemma ‘on the nose’ (cf. [20]). Let us point this out in the context of ideals of linear type.

Proposition 3.13. Let J ⊂ I be ideals of the ring R. Assume that I and J/I are ideals oflinear type (with respect to R and R/J respectively). Then J ∩ Im = JIm−1 for m > 0.

Proof. Consider the diagram of symmetric algebras and Rees rings associated to the idealsJ and I :

0 // J · Sym(I) //

��

Sym(I)+ //

��

Sym(I/J)+ //

��

0

0 // JtR(I) // R(I)+ // R(I/J)+ // 0

.

From the assumption, the lower sequence is exact. But this means that for eachm, (I/J)m =Im + J/J ' Im/JIm−1, and the assertion follows.

4 Approximation complexes

In this section we give the construction of a double complex associated to a sequence ofelements in a ring R. It bears a strong similarity to the ordinary Koszul complex but itsadded usefulness lies on the points where the two complexes differ.

12

Let us recall the definition of the graded Koszul complex associated to a map of R-module θ : G → B. For a module M over the commutative ring R, we shall denote theexterior and symmetric algebras of M respectively by

∧M and Sym(M). Their components

of degree t will be indicated by∧tM , and, in the case of the symmetric algebra, by either

Symt(M) or St(M).Let

Gψ //

ϕ

��

B

R

be a diagram of R-modules and homomorphism. We shall make the algebra∧G⊗RSym(M)

into a double complex.

dϕ = ∂ :r∧G⊗ St(B)→

r−1∧G⊗ St(B)

is the usual graded Koszul complex with coefficients in Sym(B) :

∂(e1 ∧ . . . ∧ er ⊗ w) =∑

(−1)i−1e1 ∧ . . . ∧ ei ∧ . . . ∧ er ⊗ ϕ(ei)w.

The other differential is:

dψ = ∂′ :

r∧G⊗ St(B)→

r−1∧G⊗ St+1(B)

∂′(e1 ∧ . . . ∧ er ⊗ w) =∑

(−1)i−1e1 ∧ . . . ∧ ei ∧ . . . ∧ er ⊗ ψ(ei) · w.

where ψ(ei) · w indicates the product of ψ(ei) and w in Sym(B).Straightforward verification shows that ∂ and ∂′ (skew) commute. The resulting differ-

ential graded algebra will denote by L = L(ϕ,ψ).Let x = {x1, . . . , xn} be a generating set of the ideal I, and let ϕ : Rn → R be the

mapping defined by the matrix [x1, . . . , xn].

Definition 4.1. L = L(ϕ, identity) will be called the double Koszul complex associatedto the sequence x.

To make the next observations apparent, we display a portion of the complex L :

∧r+2(Rn)⊗ St−1(Rn) //

��

∧r+1(Rn)⊗ St(Rn) //

��

∧r(Rn)⊗ St+1(Rn)

��∧r+1(Rn)⊗ St−1(Rn) //

��

∧r(Rn)⊗ St(Rn) //

��

∧r−1Rn ⊗ St+1(Rn)

��∧r(Rn)⊗ St−1(Rn) //∧r−1(Rn)⊗ St(Rn) //

∧r−2Rn ⊗ St+1(Rn)

We note that the single complex L(∂) is simply the ordinary complex associated to thesequence x in the polynomial ring S = Sym(Rn) ' R[T1, . . . , Tn]. As for the ‘horizontal’

13

complex L(∂′), it is also an ordinary Koszul complex, but constructed over the sequenceT = {T1, . . . , Tn}. Thus we have a grading of L(∂′) by the subcomplexes of R-modules,L =

∑Lt

Lt =∑r+s=t

r∧(Rn)⊗ Syms(R

n)

and the Lt are exact for t > 0. This observation plays a key role in the sequel.This construction may be extended by attaching coefficients form an R-module. Thus to

the sequence x and the R-module M there is the associated double complex L(x;M) = M⊗∧(Rn)⊗Sym(Rn), obtained by tensoring the previous complex by M. The differentials ∂ and

∂′ are, respectively, obtained by viewing L(x;M) of the Koszul complexes, KS(x;M ⊗R S)and KS(T;M ⊗R S), respectively.

If we denote by Z = Z(x;M) the submodule of cycles of K(x;M), by B = B(x;M) itsboundaries and by H = H(x;M) its homology, the commutativity of ∂ and ∂′ allows toderive from the complex L(x;M) several other complexes:

Z = Z(x;M) = {Z ⊗ S, ∂′}

B = B(x;M) = {B ⊗ S, ∂′}

and

M =M(x;M) = {H ⊗ S, ∂′}.

These complexes are called the approximation complexes associated with x and M.We shall focus on the complexes M and Z.

The M-complex, as well as the Z-complex, are graded complexes over the polynomialring S. The t-th homogeneous part,Mt ofM is a complex of finitely generated R-modulesof the following form:

Mt : 0→ Hn ⊗ St−n∂′→ . . .

∂′→ H1 ⊗ St−1∂′→ H0 ⊗ St → 0.

In a similar manner Zt is a complex of finitely generated R-modules. For certain useshowever we must viewM and Z as complexes over the polynomial ring S. Thus, for instance,the Z-complex may be written:

Z : 0→ Zn ⊗ S[−n]∂′→ . . .

∂′→ Z1 ⊗ S[−1]∂′→ Z0 ⊗ S → 0,

where, as usual, S[k] denotes the graded S-module with Sk+m for its component in degreem. In this fashion all the maps in these complexes have degree 0.

There are still other complexes associated to L(x;M). Thus if we denote by Z∗(x;M)the cycles of K(x;M) with coefficients in (x)M, we see that Z∗(x;M) is a subcomplex ofZ(x;M) and B(x,M) ⊂ Z∗(x;M). The module H∗ = H∗(x;M) = Z(x;M)/Z∗(x;M) willbe referred to as the syzygy part of H (cf. [42]). finally, the quotient Z∗(x;M)/B(x,M)will be denoted by δ(x;M).

δ(x;M) will be one of our most explicit syzygetic measures. In degree 1 and for M = R,note that δ(x;R)1 is just δ1(x) defined in §2.

14

The corresponding complexes induced by ∂′ will be denoted:

M∗ =M∗(x;M) = {H∗ ⊗ S, ∂′},

and∆ = ∆(x,M) = {δ(x;M)⊗ S, ∂′}.

These complexes made their appearance in [43], [44], [18] and [19], to where we shalloccasionally refer for some of their properties, especially those that mimic the properties ofthe ordinary Koszul complexes. We shall now remark on their difference.

While the complexes defined above depend on the sequence x and the module M, when-ever the context allows we shall adopt the simpler notation of M, Z, etc. Also, we shallwrite several proofs for M = R, since the general case is similar.

We first consider the dependence ofM and Z on the generating set {x1, . . . , xn} of theideal I = (x). For this purpose we recall two elementary properties of the graded Koszulcomplexes associated to a map ϕ : G→ B.

Let ϕ′ : G′ → B′, G = G′ ⊕R, B = B′ ⊕R and put ϕ = ϕ′ ⊕ id : G→ B. denote by L′and L the corresponding complexes. Note that L′ is a subcomplex of L.

Proposition 4.2. H•(L′) = H•(L).

This is the “cancelling by R” property of L. It is the consequence of an underlyingmapping cylinder construction in L.

Let now ϕ′ : G′ → B and define ϕ : G = G′ ⊕ R → B, ϕ = ϕ′ + 0. Denote again by L′and L the corresponding complexes.

Proposition 4.3. H•(L) = H(L′)⊗R[z], where degree(z) = 1 and z2 = 0.

This is immediate from the definition of L.Assume now G = G′⊕Re = Rn−1⊕Re, ϕ(e) = 0, Re ' R. Let L′ be the (sub-)complex

determined by G′ and the restriction of ϕ.

Proposition 4.4. H•(L′) = H•(L).

Proof. According to the definition of M, its component Mr,t is given by

Hr(K)⊗ St(B) = (Hr(K′)⊕Hr−1(K′)⊗Re)⊗

t∑j=0

Sj(B′)et−j

which we rearrange as the sum of three subcomplexes:

[Hr(K′)⊗ St(B′)] ⊕ [Hr(K′)⊗ (

∑t−1j=0 Sj(B

′)et−j)] ⊕ [Hr−1(K′)⊗Re⊗ (∑t

j=0 Sj(B′)et−j)]

M′r,t ⊕ Ar,t ⊕ Br,t

Note that Ar,t ' Br+1,t−1 induced by multiplication by e, on the first factor. It is clear thatthe complex

Cd = {Ar,t ⊕ Br,t | r + t = d}

is a mapping cylinder of the same kind that underlies 4.2. As a consequence we have thedesired isomorphism.

15

A corollary is that the Hr(M) ’s not depend on ϕ. Indeed, if ϕ : G = Rn−1 ⊕ Re andϕ(e) ∈ ϕ(Rn−1), we may easily arrange a change of basis so that G = Rn−1 ⊗Re′ but nowϕ(e′) = 0. (This shows the role of ψ = identity, and we did not have to take it into account).Since any two presentations of I, ϕ : Rn → R and θ : Rm → R, may be compared, in thesense above, to the sum map.

Similarly the homology of the complex Z is independent of the generating set x.

Note that while the homology of the ordinary Koszul complexes depends on the gener-ating sets of the ideal it does have certain properties - rigidity, among others - not sharedby the present complexes (cf. §13).

We now begin pointing out the reasons for the usefulness of the complexes Z and M.

First, we claim that H0(Z(I;R)) = SymR(I). Indeed, looking at the presentation of Iprovided by

0→ Z1(K)→ Rn → I → 0

we obtain

Z1(K)⊗ Sym(Rn)→ Z0(K)⊗ Sym(Rn) = Sym(Rn)→ H0(Z(I;R))→ 0,

the assertion follows from the universal properties of symmetric algebras.

Similarly, one sees that H0(M(I;R)) = SymA(I/I2), A = R/I. (As a notation, weshall write SymA(I/I2) = Sym(I/I2).) It is also easy to see that H•(Z(I;R)) (resp.H•(M(I;R))) is a finitely generated Sym(I) (resp. Sym(I/I2))-module.

For M 6= R, the meaning of H0(Z(I;R)) and H0(M(I;R)) is not so clear-cut. Thereare however natural surjections.

α : H0(Z(I;R))→ R(I;M) =⊕j

IjM = associated Rees module,

and

β : H0(M(I;R))→ grI(M) =⊕j

IjM/Ij+1M = associated graded module.

α and β are natural identifications in degree 0.

We now point out a basic relationship between the complexes Z = Z(x;M) and M =M(x;M)

Proposition 4.5. For each positive integer t there exists an exact sequence of R-modules:

. . .→ Hr(Zt+1)→ Hr(Zt)→ Hr(Mt)→ Hr−1(Zt+1)→ . . .

Proof. Consider the defining exact sequences

0→ Zt → Lt → Bt−1[−1]→ 0

and

0→ Bt → Zt →Mt.

16

As observed earlier, L(∂′) is the Koszul complex associated to the indeterminates of thering R[T1, . . . , Tn] and the module M ⊗R[T1, . . . , Tn]. Thus

Hr(Lt) =

{M for r = t = 00 otherwise

(1)

When this is taken into the two long homology exact sequences, the desired exactnessensues.

Of interest here will be the tail of this exact sequence, that is

H1(Mt)σ→ H0(Zt+1) = St+1(I)

λ→ H0(Zt) = St(I) → H0(Mt) = St(I/I2) → 0

where we denote by σ (or σt) and λ (or λt) the connecting homomorphism and the down-grading homomorphism, respectively. Tracing through the meaning of the complex Lt, itis easy to verify that λt is just the downgrading mapping from St+1(I) to St(I) that wasconsidered in §3.

Corollary 4.6. The following are equivalent for the complexes M and Z.

(a) M is acyclic.

(b) Z is acyclic and λ is injective.

Proof. Since Hr(Z0) = 0 for r > 0, the assertion follows by induction and the propositionabove

Taking into account the exact sequence of (4.5), one has the following statement aboutthe mappings σ, λ and α. It will be our key technical device in checking whether an idealis of linear type.

Theorem 4.7. Let I be an ideal and M an R-module. Let Z and M be the complexesconstructed on a generating set for I. The following are equivalent:

(a) σ is the zero mapping.

(b) λ is injective.

(c) α is an isomorphism.

Proof. Consider the commutative exact diagram

H1(Mt)σ // H0(Zt+1)

λ //

αt+1

��

H0(Zt) //

αt

��

H0(Mt) //

βt��

0

0 // It+1M // ItM // ItM/It+1M // 0

Since α and β are isomorphism in degree 0, the assertion follows by induction and theexactness of the diagram.

Corollary 4.8. Let I be an ideal of the ring R. If the complex M(I;R) is acyclic, then Iis of linear type.

17

One may extend the definition of ideals of linear type (relative to R) to coefficients inmodules as well.

Let I be an ideal of the ring R and let M be an R-module.

Definition 4.9. I is of linear type with respect to M if the mapping α : H0(Z(I;M)) →R(I;M) is an isomorphism.

The argument of (3.1) carries over to this setting, and we have

Proposition 4.10. Let R be a Noetherian ring and let M be a finitely generated module.Then α is an isomorphism if and only if β is an isomorphism.

5 Depth and acyclicity

In this section we consider exactness in the complexes M and Z. For simplicity we shallassume from now on that R is a Cohen–Macaulay ring. It allows us to state the next resultswith “heights”, which is more convenient.

In the sequel we shall denote the minimal number of generators of a module E by ν(E),and the height of an ideal I by ht(I).

Theorem 5.1. Let R be a Cohen–Macaulay ring and let I be an ideal of height ` generatedby n = `+ s elements. Assume:

(a) For each prime ideal P ⊃ I, ν(IP ) ≤ htP.

(b) For each integer r and each prime P ⊃ I, depth (Hr)P ≥ inf{ht(P/I), r}.

where the Hr denote the homology modules of the Koszul complex on the chosen n generatorsof I.

Then the complex M(I;R) is acyclic.

The proof (cf. [18]) uses the following combination of the acyclicity lemma of [34] and[25, p. 103]

Proposition 5.2. Let R be a local ring and let

C : 0→ Cs → . . .→ C1 → C0 → 0

be an complex of finitely generated R-modules.

(a) If (i) depth (Cj) ≥ j and (ii) depth (Hj(C)) = 0 or Hj(C) = 0 for j ≥ 1, then C isacyclic.

(b) Moreover, if for some positive integer t, depth Cj ≥ t + j for each j ≥ 0, thendepth H0(C) ≥ t.

Remark 5.3. Note that the condition (a) in (5.1) (cf. 4.7 and 2.4) is necessary for theacyclicity of M(I;R). The formulation of (b) is recognizably cumbersome. Nevertheless, itis, due to 4.2 and 4.3, independent of the chosen generating set of n generators. Later weconsider and the local behavior of the depths in the Koszul homology.

18

Proof. (of 5.1, cf. [44]) We may assume that R is a local ring and that I is minimallygenerated by ` + s elements, ` =grade(I), s ≥ 1. (Recall that grade may increase underlocalizations, but this just improves the hypotheses).

Consider the complex

Mt : 0→ Hn ⊗ St−n → . . .→ H1 ⊗ St−1 → H0 ⊗ St → 0

These complexes have length at most s, since Hr = 0 for r > s. We thus have complexesof type

0→ Ls → . . .→ L1 → L0 → 0

where, for a given r, Lr = Hr ⊗ St−r.By (b) depth(Lr) ≥ inf{dim(R/I), r} ≥ r, and we may appeal to 5.2 to complete the

proof.Because 4.6, the Z-complex is also acyclic under the conditions of 5.1. The following is

however a slightly improved version.

Theorem 5.4. Let R be a Cohen–Macaulay ring and let I be an ideal of height ` ≥ 1generated by n = `+ s elements. Assume:

(a) For each prime P ⊃ I, ν(IP ) ≤ 1 + ht(P ).

(b) For each integer r and each prime P ⊃ I depth(Hr)P ≥ inf{ht(P/I), r} − 1, wherethe Hr denote the homology modules of the Koszul complex on the chosen generatorsof I.

Then the complex Z(I;R) is acyclic.

The proof is similar, with the hypothesis ` ≥ 1 making for Zn = 0, which will shortenthe complexes to which 5.2 will apply.

We shall now illustrate the uses of these complexes for ideals that are sufficiently close tocomplete intersections. Although not strictly necessary, we shall assume that R is a Cohen–Macaulay local ring. Let I be an ideal of height ` minimally generated by ν(I) = ` + selements; we discuss the cases s ≤ 3.

(i) s = 0 : In this case both complexes are exact, and grI(R) = Sym(I/I2) is isomorphicto the polynomial ring (R/I)[T1, . . . , T`]. On the other hand, the dimension and depthof Sym(I) can readily be determined from the acyclicity of the Z-complex - in fact,Sym(I) is Cohen–Macaulay along with R. These facts will however be taken up, withgreater generality, in §9.

(ii) s = 1: By 5.4 the Z-complex is always acyclic. The M-complex will be exact if foreach minimal prime P ⊃ I, IP is generated by a regular sequence. Indeed, it suffices,by 5.1, to show that if R is local and s = 1, then depth(H1) ≥ 1. More generally:

Proposition 5.5. ([3], [30]) Let R be a Cohen–Macaulay local ring and let I be an idealgenerated by 1 + ht(I) elements. Then

depth(H1) ≥ inf{dim(R/I), 2 + depth (R/I)},

with equality if R is Gorenstein.

19

This situation provides examples of non-Cohen–Macaulay ideals I, for with both Sym(I/I2)and Sym(I) are Cohen–Macaulay (cf. [44] and [19]).

(iii) s = 2: Assume now that R is a Gorensntein local ring and that I is a Cohen–Macaulayideal. The depth of the Koszul homology modules of such ideals was determined in[2]:

Proposition 5.6. Let R be a Gorenstein local ring and let I be a Cohen–Macaulay idealminimally generated by 2 + ht(I) elements. Then the Koszul homology modules Hi areCohen–Macaulay (R/I)-modules.

Proof. Let K• be the Koszul complex on a minimal generating set of I. Since R is assumedto be Gorenstein ring, the last non-vanishing homology module of K•, H2, is the canonicalmodule of R/I, and therefore a Cohen–Macaulay module. We thus only have to show thatH1 is Cohen-Macaulay. Let x ∈ R/I be a regular element and let y ∈ R be an R-regularelement which is a lifting of x. From the short exact sequence

0→ K•y→ K• → K ′• → 0

we obtain the long exact sequence

0→ H2x→ H2 → H2(K

′•)→ H1

x→ H1 → H1(K′•)→ H0

x→ H0 → H0(K′•)→ 0

In the above sequence the mapping H2 → H2(K′•) is surjective, and the mapping H0

x→H0 is injective. Therefore x is H1-regular and H1/xH1 = H1(K

′•). By induction on the

dimension of R/I, the assertion follows.

In this case (s = 2) we have (cf 5.1 and 5.4):

(iiia) If ν(IP ) ≤ ht(P ) for each prime P ⊃ I, then the M-complex is acyclic.

(iiib) If ν(IP ) ≤ 1 + ht(P ) for each prime P ⊃ I, then the Z-complex is acyclic.

(iv) s = 3: Let R be a Gorenstein local ring and let I be a Cohen–Macaulay ideal generatedby ν(I) = 3 + ht(I) elements. As in 5.6, considering the Koszul complex associated toa minimal generating set of I, we conclude that H3 is Cohen–Macaulay. Here howeverthe depths of H1 and H2 are not known - much less whether they are Cohen–Macaulay.

(iva) If, in the situation above, for every prime ideal P ⊃ I, ν(IP ) ≤ ht(P ), then theZ-complex is acyclic.

Proof. Let us apply 5.2 directly to the Z-complex (n = `+ 3):

Zt : 0→ Zn ⊗ St−n → . . .→ Z2 ⊗ St−2 → Z1 ⊗ St−1 → Z0 ⊗ St → 0.

Since we may assume ` ≥ 1, Zn = 0 and the complex Zt has length at most ` + 2. If Kdenotes the corresponding Koszul complex, for r ≥ 4 the truncated complex

0→ K`+3 → . . .→ Kr+1 → Br → 0

20

provides a minimal projective resolution of Br (= Zr) and thus, by Auslander-Buchsbaumequality,

depth(Zr) = dimR− (`+ 3− r − 1) = (dimR− `− 2) + r ≥ r + 1

since ν(I) = `+ 3 ≤ dimR.

For r = 3 : From the exact sequence

0→ Br → Z3 → H3 → 0

we get depth Z3 ≥ 3, as H3 is Cohen–Macaulay of depth equal to dim(R/I) ≥ 3, anddepth B3 ≥ 3, by the truncated Koszul complex (which it is still exact for r = 3).

On the other hand, Z1 and Z2 are second-syzygy modules, and thus depth Z2 ≥ 2 anddepth Z1 ≥ 2. (In fact, Z1 has a much higher depth).

The acyclicity of the Z-complex now follows from a direct application of 5.4.

It is easy to see that the depth of the irrelevant prime ideal of Sym(I) is at least `+ 3.

(ivb) As for the M-complex, its exactness is a bit harder to deal with since the use of 5.1would require depth H2 ≥ 2, and depth H1 ≥ 1. An alternative would be to showdirectly the isomorphism of the blowing-up rings Sym(I) and R(I), and then makeuse of the preceding and 4.6.

6 Sequences

The aim of this section is a discussion of certain generalizations of the notion of a regularsequence that play, in the setting of the approximation complexes, a role comparable tothat of regular sequence for ordinary Koszul complexes.

A key notion is that of a d-sequence - already briefly discussed in section 2 in the contextof ideals of linear type - with will assume a dominant role in sections 12 e 13. We show thatthe existence of ‘straight-up’ d-sequences are ubiquitous in Noetherian rings and use it toprove a coarse one-step Cohen–Macaulayfication.

6.1 Generalizations of regular sequences

We begin with a listing of the definitions of the various kinds of sequences.

Definition 6.1. Suppose x = {x1, . . . , xn} is a sequence of elements in a ring R and M isan R-module. The sequence x is called a:

(a) d-sequence with respect to M, if:

(a1) x is minimal generating set of the ideal I = (x) = (x1, . . . , xn).

(a2) (x1, . . . , xi)M :M xi+1xk = (x1, . . . , xi)M :M xk, for i = 0, . . . , n − 1, and k ≥i+ 1.

(b) relative regular sequence with respect to M, if (x1, . . . , xi)IM :M xi+1 ∩ IM =(x1, . . . , xi)M for i = 0, . . . , n− 1, where I = (x).

21

(b∗) relative∗ regular sequence with respect to M, if (x1, . . . , xi)M :M xi+1 ∩ IM =(x1, . . . , xi)M for i = 0, . . . , n− 1, where I = (x).

(c) proper sequence with respect to M, if xi+1Hj(x1, . . . , xi;M) = 0 for i = 0, . . . , n−1,where Hj(x1, . . . , xi;M) denotes the Koszul homology associated to the initial subse-quence {x1, . . . , xi}.

The notion (a) was introduced by Huneke ([20]), who has constructed many interestingclasses of examples and proved some of its fundamental properties. Both (b) and (b∗) wereintroduced by Fiorentini ([11]). As for (c), it was first defined in [18] for its usefulness in ex-amining exactness in the approximation complexes (see §12). There are also unconditionedversions of each of these notions - that is, independent of reorderings - but they will not bediscussed here. Finally, in the enlarged context here - that of R-modules - the minimalitycondition (a1) above does not play a major role. For this reason we sometimes refer to thefollowing condition - equivalent to (a2) - as a d-sequence also (or, a d∗-sequence when adistinction is needed):

(a∗2) (x1, . . . , xi)M :M xi+1 ∩ IM = (x1, . . . , xi)M for i = 0, . . . , n− 1, where I = (x).

Several implications hold between these kinds of sequences. We indicate in the diagrambelow those we shall be interested in:

d− sequence⇓

d∗ − sequencem

relative∗ regular sequence⇓

relative regular sequence⇓

proper sequence

The implication d∗-sequence ⇒ relative∗ regular sequence follows from 3.7. For theconverse (cf. [18]), let a ∈ (x1, . . . , xi)M : xi+1xk; then axk ∈ (x1, . . . , xi)M : xi+1 ∩ IM =(x1, . . . , xi)M and thus a ∈ (x1, . . . , xi)M : xk.

Fiorentini (cf. [11]) showed that for a relative regular sequence x = {x1, . . . , xn} onehas

Zj(x1, . . . , xi;M) ∩ IK = Bj(x1, . . . , xi;M)

for i = 0, . . . , n− 1 and j > 0, where K denotes the Koszul complex on the full sequence x.The last implication, relative regular sequence ⇒ proper sequence, then follows.

(i) for M = r, d-sequences show features common to regular sequences and generatingsets of projective ideals. Thus, for instance,

(ia) If dimR = r, then every d-sequence has at most r + 1 elements.

(ib) For a Dedekind domain every ideal is generated by a d-sequence.

(ii) For other modules however the bound (ia) ceases to exist, as when (R,m) is a localring and M = R/m.

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There is another way to express condition (c) above. Using the recursive form of theKoszul homology modules (cf. [29]), (c) may also be written:

(c∗) x is a proper sequence with respect to M , if IHj(x1, . . . , xi;M) = 0 for i =0, . . . , n− 1, j > 0, and Hj(x1, . . . , xi;M) with the earlier meaning.

Contrary to d-sequences for M = R, there may not exist dimension dependent boundsfor the length of (minimally generated) proper sequences. Thus, if (R,m) is an Artinianlocal ring and m2 = 0, then m is generated by a proper sequence.

One may to, possibly, get regular sequences from d-sequences or proper sequencesemerges directtly from the definition (c∗): Passing to the localization Rxn it follows thatH1(x1, . . . , xn−1;M)xn = 0. Thus, if (x1, . . . , xn−1)Mxn 6= Mxn , then one obtains a regularMxn-sequence of length n− 1 in the ideal (x1, . . . , xn)Rxn .

Before we make use of these remarks, let us show that d-sequences - especially for certainstraight-up types - are rather ubiquitous. For simplicity we consider the case of M = Rsince the extension to arbitrary modules is immediate.

Proposition 6.2. Let I be an ideal of height n of the Noetherian ring R. Then I containsa d-sequence (with respect to R) of length (and height) n.

Proof. Let {P1, . . . , Pr} be the minimal prime ideals of I. We shall use the followingnotation. For an R-module M, denote by Z(M) the union of the associated primes of Mwhich are properly contained in one of the Pi.

We may assume n > 0. Let y1 ∈ I \ Z(R), and stabilize it, that is, pick m large enoughso that 0 : ym1 = 0 : ym+1

1 . Put x1 = ym1 ; if n = 1, {x1} will do.If n > 1, consider the ideal J1 = 0 : (0 : x1). We claim that J1 is not contained in any

prime ideal associated to R and properly contained in some Pi. Otherwise, for such a primeP we would have: if x1 is regular in RP , 0 : x1 = 0, with is impossible; if x1 is not regularin RP , P contains some associated prime of R, which conflicts with the choice of y1 - andhence of x1. Thus we may pick y2 ∈ I ∩J1 \{Z(R)∪Z(R/(x1))}} and stabilize it - meaningnow take m lawge enough so that 0 : ym2 = 0 : ym+1

2 and (x1) : ym2 = (x1) : ym+12 . Put

x2 = ym2 .In general, if x1, . . . , xs, have been chosen and n > s, one picks xs+1 in the following

manner. Again it is easy to see that Js = (x1, . . . , xs−1) : ((x1, . . . , xs−1) : xs) is notcontained in any prime properly inside one of the Pi. Now pick ys+1 in I ∩ J1 ∩ . . . ∩Js \ {Z(R) ∪ Z(R/(x1)) ∪ . . . ∪ Z(R/(x1, . . . , xs))}, and stabilize it. This means picking mlarge enough so that for each initial subsequence {x1, . . . , x`}, ` ≤ s, (x1, . . . , x`) : yms+1 =(x1, . . . , x`) : ym+1

s+1 . Finally put xs+1 = ym+1s+1 . Finally put xs+1 = yms+1.

We claim that {x1, . . . , xn} is a d-sequence. Since the chosen elements will generatean ideal of height n, the minimality condition is ensured. Note also that any sequence{x1, . . . , xs, xn}, s ≤ n − 1, is , by induction, a d-sequence since it contains at mostn − 1 elements and the construction leading to it is the same that leads to the sequence{x1, . . . , xs, xs+1}. Therefore the only step to check is the equality

(x1, . . . , xn−2) : xn−1xn = (x1, . . . , xn−2) : xn.

Let t be an element in the first ideal; then txn ∈ (x1, . . . , xn−1) : xn and since xn ∈ Jn−1,we have tx2n ∈ (x1, . . . , xn−2). The assertion now follows from the stabilization step.

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Remark 6.3. A similar construction was used by Bass ([4]) to find bounds for variousfinitistic dimensions of Noetherian rings.

6.2 Syzygetic ideals

Among the notions of syzygetic ideals used in the past, we single out:

(a) I is syzygetic1 if δ1(I;R) = δ1(I) = 0.

(b) I is syzygetic2 if δt(I) = 0 for t ≥ 1 (cf §4). This means that for some set x ofgenerators of I, Zt(x) ∩ IK = Bt(x), t > 0. (It will be the case that this definitiondoes not depend on the generating set.)

(c) I is syzygetic3 if the associated complex M(I;R) is acyclic.

Proposition 6.4. (c)⇒ (b)⇒ (a). Furthermore, none of the inverse implications is valid.

Proof. To show (c)⇒(b) we go back to the definition of the approximation complexM∗(I;R) (cf. §4). The exact sequence

0→ ∆→M→M∗ → 0

has, in degree t, the following tail

0 // δt(x) //

��

Ht

��

// H∗t //

��

0

0 // δt−1(x)⊗ S1 // Ht−1 ⊗ S1 // H∗t−1 ⊗ S1 // 0

From this diagram, we see that δ1(x) = H1(M1). Assume that we have proved δt−1(x) =0 for t ≥ 2. Since the diagram has exact rows and the middle vertical complex is acyclic, itfollows that δt(x) = 0 also.

Note also that the same argument shows that if δt−1(x) = 0, then δt(x) = ker (Ht(Mt)→Ht(M∗)). Since δ1(x) is independent of x, it follows that if the δj(x), j < t vanish, thenδt(x) will be independent of the generating set.

Remark 6.5. In §12 it will be proved that for proper sequences the complex Z(I;R) isacyclic. Thus, by 4.5, Hj(M) = 0 for j > 1. From the argument above it then follows thatfor ideals generated by proper sequences we have syzygetic1 ⇒ syzygetic2.

Let us now consider the inverse implications.

Example 6.6. Let P be a prime ideal of R = k[[x, y, z]] needing at least 5 generators; e.g.pick P to be one of the so-called Macaulay primes. According to §7, the homology modulesof the complex K are Cohen–Macaulay, meaning here they are torsion-free R/P -modules.Since P is generically a complete intersection, (H)P = (H∗)P , and thus δt(P ) = 0 for t > 0.We claim that M(P ;R) is not acyclic, Otherwise it would lead to the equality of the Reesand symmetric algebras of P and this would contradict 2.4. Thus P is syzygetic2 but notsyzygetic3.

Example 6.7. Let R = k[u, v, x, y], with the relations ux = vy, uy = vx = u2 = uv =v2 = 0, k =field. R is the symmetric algebra of a module over k[u, v]. Let I = (x, y). SinceI is the augmentation ideal of R, it follows by 2.3 that I is of linear type - in particularδ1(I) = 0. Note however that H2 = Rux = Z2 = δ2(I), and thus I is not syzygetic2.

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6.3 Coarse Cohen-Macaulayfication

We shall now make use of 6.2 to obtain a crude form of Cohen–Macaulayfication. For a ringR, by the Cohen–Macaulayfication of Spec(R) we mean a pair {X, f} of a Cohen–Macaulayscheme X and a proper mapping f : X → Spec(R) which is an isomorphism at the Cohen-Macaulay points of Spec(R). Some special - but still hard -cases have been dealt with byBrodmann and Faltings (see [7], [10]), and we shall consider here another approach to oneof its facets.

Theorem 6.8. Let (R,m) be a Noetherian local ring of dimension n, and let x ={x1, . . . , xn−1, xn = z} be a system of parameters which is a d-sequence (such systems al-ways exist by 6.2). Let S be the subring of Rz generated by R and the xi/z, i < n, and let Pbe the prime ideal (m, x1/z, . . . , xn−1/z). Then SP is a Cohen–Macaulay ring of dimensionn.

Proof. Write I = (x) and let C be the Rees algebra C = R[It] =∑

j Ijtj . Denote the

elements x1t, . . . , xn−1t, xnt = zt respectively by ζ1, . . . , ζn−1, ζ. Note that in Cζ the elementxn is regular and Cζ/(xn) = grI(R)ζ .

Since {x1, . . . , xn} is a d-sequence, by [22] (see ?? for a converse and fuller discussion)the corresponding 1-forms {ζ1, . . . , ζn−1, ζn = ζ} in grI(R) is also a d-sequence. Thus, by?? and the choice of the xi’s, {ζ1, . . . , ζn−1} form a regular sequence in grI(R)ζ . Altogetherwe have that {xn, ζ1, . . . , ζn−1} is a regular sequence in Cζ .

Finally, as Cζ = S[ζ, ζ−1] - S is the 0-th homogeneous piece of Cζ - the rest of theassertion follows from standard facts on the dimension of blowing-up rings.

7 Depth of the Koszul homology modules

The applications in §5 and, kore interestingly, in §9, of the approximation complexes toderive arithmetical properties of the blowing-up rings require detailed knowledge of thedepths of the homology modules Hi(I;R). In this section we first show how much simplerit is to estimate those depths for ideals which are generated by proper sequences with highresidual depths. We then prove a result of Huneke ([24]) on the behaviour of the Cohen–Macaulayness of the Hi(I;R) under linkage. This allows us to organize around it severalearlier known cases of ideals with Cohen–Macaulay Koszul homology, e.g. perfect ideals ofheight 2 and Gorenstein ideals of height 3.

7.1 Ideals generated by proper sequences

Let x = {x1, . . . , xn} be a proper sequence with respect to the ring R itself. One has thefollowing exact sequence, derived from the definition (cf. §6):

0→ H1(x1)→ Rx1→ R→ R/(x1)→ 0

0→ H1(x1, . . . , xi)→ H1(x1, . . . , xi+1)→ Rixi+1→ Ri → Ri+1 → 0

where Ri = R/(x1, . . . , xi), and, for j > 1,

0→ Hj(x1, . . . , xi)→ Hj(x1, . . . , xi+1)→ Hj−1(x1, . . . , xi)→ 0.

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From these sequences we can obtain depth estimates for Hj(I;R) in terms ofdepth(R/(x1, . . . , xi)). The following is an extreme case.

Proposition 7.1. Let I be an ideal generated by a proper sequence x = {x1, . . . , xn}.Assume:

(a) depth(R/(x1, . . . , xi)) ≥ depth R− i, 1 ≤ i ≤ n− 1.

(b) depth R/I ≥ depth R− n+ i, 1 ≤ i ≤ n.

Proof. It is immediate from depth-chasing along the exact sequences above.

Remark 7.2. (i) One instance where the residual depth conditions above are satisfiedis that of monomial ideals. In fact, if I = (x1, . . . , xn), where the xi’ s are monomi-als in some set of indeterminates, it follows from [?] that the projective dimension ofR/(x1, . . . , xi) is at most i. Thus, by the Auslander-Buchsbaum equality ([29]), one hasthat depth R/(x1, . . . , xi) is, locally, at least depth R − i. In [18] necessary and sufficientconditions where given for a sequence of monomials to be a d-sequence. For ‘square-free’monomials, Kuhl ([27]) has shown that proper sequences are the same as d-sequence.

(ii) Another case to which 7.1 would apply is that of an almost complete intersection,already treated in 5.5.

7.2 Koszul homology and linkage

We shall now discuss a theorem of Huneke ([24]), and some of its consequences.The notion of ‘linkage’ that we shall deal with is the following ([35]). Let R be a Cohen–

Macaulay local ring and let I and J be two ideals of R. I and J are said to be linked isthere exists a regular sequence x = {x1, . . . , xn} in I ∩ J such that I = (x) :R J andJ = (x) :R I. (Notation: I ∼(x) J, or simply I ∼ J). Note that the definition impliesAss(R/I) ⊂ Ass(R/(x)), so that I (and also J) is height unmixed.

The linkage class of the ideal I, L(I), consists of all ideals J for which there exists asequence of linkages I ∼ L1 ∼ L2 ∼ . . . ∼ Ln ∼ J. Of interest here will be the subset Le(I)of ideals linked to I through an even number of linkages.

Several important linkage classes have been studied:

Theorem 7.3. ([?],[26]) Let R be a Gorenstein local ring and assume I ∼ J. Then I is aGorenstein ideal if and only if J is an almost complete intersection.

Theorem 7.4. ([35]) Let R be a regular local ring and let I be a Cohen–Macaulay ideal ofheight two. Then I is in the linkage class of a complete intersection.

Theorem 7.5. ([?], [49])Let R be a regular local ring and let I be a Gorenstein ideal ofheight three. Then I is in the linkage class of a complete intersection.

Note that in these two cases, it follows from 7.3 that I is even linked to either a completeintersection or to an almost complete intersection.

A basic property on the behaviour of the Cohen–Macaulay character under linkage isthe following:

Theorem 7.6. (Dubreil’s Theorem) (cf. [35]) Let R be a Gorenstein local ring and let I bea Cohen–Macaulay ideal. If J is linked to I, then J is a Cohen–Macaulay ideal.

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Otherwise stated, If R is a Gorenstein local ring, then the Cohen–Macaulayness ofH0(I;R) is preserved under linkage. Unfortunately the higher Koszul homology modules donot always show this behaviour. The following, however, is sufficient for several applications.

Theorem 7.7. ([22]) Let R be a Cohen–Macaulay local ring and suppose L is an ideal ofgrade n. Let x = {x1, . . . , xn} and y = {y1, . . . , yn} be two regular sequences in L and setI = (x) : L and J = (y) : L. If Hi(I;R) are Cohen–Macaulay for 0 ≤ i ≤ m, then Hi(J ;R)are Cohen–Macaulay for 0 ≤ i ≤ m.

The assertion is to the effect that the Cohen–Macaulayness of the Koszul homology of Ibehaves well under even linkage. There are examples ([24]) showing that this is not alwaysso under linkage in general.

From 7.1 it follows that if R is Cohen–Macaulay and I is a Cohen–Macaulay ideal thatis an almost complete intersection, then Hi(I;R) are Cohen–Macaulay (only i = 0, 1 needbe considered). Taking into account the remark following ?? we have.

Corollary 7.8. Let R be a regular local ring and let I be either a Cohen–Macaulay idealof height two or a Gorenstein ideal of height three. Then the Koszul homology modulesHi(I;R) are Cohen–Macaulay.

Remark 7.9. Both cases in 7.8 had been treated, directly, in [2] and [23], respectively.

We sketch in detail the proof of 7.7. Despite its elementary nature, it displays a numberof useful techniques on dealing with the Koszul homology modules, which may be pertinentin other circumstances.

Theorem 7.10. (The dimension of the Koszul homology modules) Let R be a Cohen–Macaulay local ring of dimension d and let I be an ideal of height n. If Hi(I;R) 6= 0, thenits dimension is d− n.

Proof. Let x = {x1, . . . , xn} be a maximal regular sequence in I and let K(I;R) be theKoszul complex on a set of r generators of I. First, since the Hi(I;R) are annihilated by I,dimHi(I;R) ≤ d− n.

The first non-vanishing homology module is (cf. [?]): Hr−n(I;R) ' (x) : I/(x). Itsassociated primes ideals are minimal primes of (x), and so dimHr−n(I;R) = d− n.

To show dimHk(I;R) = d − n, 0 < k < r − n, assume otherwise and let P be anassociated prime of Hr−n(I;R). Since ht(P ) = n, localizing at P we get Hk(I;R)P = 0 andHr−n(I;R)P 6= 0, thus contradicting the rigidity of the Koszul complex.

The dependence of the Koszul homology modules on the generating set of I is containedin 4.3. Thus, if x’ is the generating set obtained by adding a superfluous element to x, say,x′ = {x1, . . . , xn, 0}, then

Hi(x′;R) = Hi(x;R)⊕Hi−1(x;R).

In particular, if the Hi(x;R) are Cohen–Macaulay for 0 ≤ i ≤ m, so will be the Hi(x;R)in the same range. Since any two generating sets for I may be thus compared to the setx ∪ y, the Cohen–Macaulay condition, in a range as above, is seen as independent of thegenerating set.

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Let I be an ideal generated by the sequence x = {x1, . . . , xn} and suppose x1 is a regularelement of R. We have then the exact sequences of Koszul complexes

0→ K(x;R)x1→ K(x;R)→ K(x;R/(x))→ 0.

The last complex is the same as K(x∗;R∗), x∗ = {0, x∗2, . . . , x∗n}, where ‘∗’ denotes thecanonical homomorphism R→ R/(x1) = R∗.

Taking the homology sequence, we obtain the exact sequence:

0→ Hi(x;R)→ Hi(x∗;R∗)→ Hi−1(x;R)→ 0.

From ?? it follows that the passage R → R∗ preserves the Cohen–Macaulay property ofHi(I;R), 0 ≤ i ≤ m.

We begin to prove 7.7. First, note that the case n = 0 is vacuous. For n = 1, considerI = (x) : L and J = (x) : L.

(a) The condition I = (x) : L implies that I ' HomR(L,R). Indeed the moduleHomR(L,R) may be identified to the set R :Q L = {r ∈ Q | rL ⊂ R}, Q = total ringof fractions of R. The mapping ϕ : I → R :Q L given by ϕ(a) = a/x is easily checked to bean isomorphism.

The hypothesis of 7.7, for n = 1, thus implies that I ' J. Denote this isomorphism byα -which is realized as multiplication by a fixed element of Q still denoted by α.

(b) First we need a criterion for the condition Hi(I;R) is Cohen–Macaulay. For that,let K(I;R) be the Koszul complex on a fixed set of generators x = {x1, . . . , xn}. We claimthat if ht(I), then Hi(I;R) are Cohen–Macaulay 0 ≤ i ≤ m, if and only if the cycles Ziand the boundaries Bi of K are maximal Cohen–Macaulay modules, for 0 ≤ i ≤ m. For theproof, it suffices to use induction on the defining sequences:

0→ Bi → Zi → Hi(I;R)→ 0

and

0→ Zi → Ki → Bi−1 → 0.

Assume then J = αI, α a regular element of Q, and let K(I;R) be the Koszul complexon the generating set x above. Denote by K′(J ;R) the Koszul complex corresponding tothe set y = {αx1, . . . , αxn} of generators of J.

We claim that Zi = Z ′i and Bi ' B′i. First, if we identify K and K′ as R-modules, wehave that Zi = Z ′i, since αd = d′ and α is a regular element of Q. We can then completethe didgram

0 // Zi // Ki// Bi // 0

0 // Z ′i// K ′i

// B′i// 0

and conclude that Bi ' B′i.Assume n > 1. To use induction on n, we first establish the following claim. Let

x = {x1, . . . , xn} and y = {y1, . . . , yn} be regular sequences in L. There exist elementsz1, . . . , zn−1 satisfying the following conditions:

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(a) zi = yi + ai, ai ∈ (yi+1, . . . , yn).

(b) {x0, x1, . . . , xk, z1, . . . , zi} is a regular sequence for 0 ≤ k ≤ n− i (here x0 = 0).

Suppose z1, . . . , zi−1 have been chosen. By assumption {x0, x1, . . . , xk, z1, . . . , zi−1} is aregular sequence for k ≤ n− (i−1) = n− i+1. Thus it suffices to chose zi as in (a) which isnot contained in any associated prime of R/(x0, . . . , xk, z1, . . . , zi−1), 0 ≤ k ≤ n− i. If P issuch a prime, then (yi, yi+1, . . . , yn) is not contained in P else (yi, . . . , yn) and grade(P ) ≥ n.It now follows that zi can be chosen satisfying (a) and (b) (cf. [25, Th. 124]).

Set Di = (x1, . . . , xn−i, z1, . . . , zi) : L for 1 ≤ i ≤ n − 1. Set D0 = (x1, . . . , xn) : Land Dn = (y1, . . . , yn) : L. Then D0 = I, and since (z1, . . . , zn) = (y1, . . . , yn), Dn = J.By induction on j, we will show that Hi(Dj ;R) are Cohen–Macaulay for 0 ≤ i ≤ m. Forj = 0 this is our assumption. Suppose we have shown Hi(Dj−1;R) are Cohen–Macaulay for0 ≤ i ≤ m. The elements x1, . . . , xn−j , z1, . . . , zj−1 form a regular sequence in Dj−1 ∩ Dj .Denote by ‘∗’ the homomorphism R → R/(x1, . . . , xn−j , z1, . . . , zj−1). By the choice ofthe zi, both {x1, . . . , xn−j , z1, . . . , zj−1, xn−j+1} and {x1, . . . , xn−j , z1, . . . , zj} are regularsequences (R = local, so that regular sequences can be re-arranged). Hence z∗j and x∗n−j+1

are regular elements in R∗. Note D∗j−1 = (x∗n−j+1) : L∗ while D∗j = (z∗j ) : L∗. Now we use?? and ?? to complete the induction.

Remark 7.11. Information on the depths of the Koszul homology modules Hi(I;R) alsoplays a role in an open suestion regarding the homology of the conormal module I/I2.Namely, let R be a local ring and let I be an ideal of finite projective dimension. In [47]and [48] the following conjecture was considered:

C1 (homological rigidity of the conormal module): the only possible values for theprojective dimension of I/I2 as an R/I-module are 0 and ∞.

This means that if pdR/I(I/I2) ≤ ∞, then I must be a complete intersection. Some

cases, involving ideals of small height, have been considered. The Koszul homology inter-venes directly in the following exact sequence of §2:

0→ δ1(I)→ H1(I;R)→ (R/I)n → I/I2 → 0.

If the conjecture holds for lower dimensions and H1(I;R) is Cohen–Macaulay then δ1(I) = 0and thus one has that pdR/I(I/I

2) = 0, or 1. But a theorem of Gulliksen (cf. [12]) rules outthe second possibility. This settles C1 for those cases where H1(I;R) is Cohen–Macaulay.

8 Dimension of symmetric algebras

In this section we consider ways of expressing the dimension of the symmetric algebra of amodule. These considerations will be enhanced in the case of an ideal, for they will mean afirst step towards checking whether the ideal is of linear type (cf. §2). Further applicationswill be given §§9 e 10.

We set-up the notation: Throughout R is a Noetherian (frequently Cohen–Macaulay)ring and E is a finitely generated R-module endowed with a (generic) rank in the sensethat E ⊗RQ is a free Q-module, where Q is the total ring of fractions of R. We will denoterk(E) the cardinality of the Q-basis of E ⊗R Q.

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Further notation will appear whenever a finite presentation

Rmϕ→ Rn → E → 0

is given:

It(ϕ) := ideal of R generated by the t× t minors ofthe matrix of ϕ

(this is independent of base choice).

rk(ϕ) := the rank ϕ = rk(Im(ϕ)) = largest r such that Ir(ϕ) 6= 0.

t0(ϕ) := inf{t ≥ 1 | grade(Is(ϕ)) ≥ rk(ϕ)− s+ 1, for, s ≥ t}.

Note that these depend on the presentation. To get absolute invariants (i.e. of themodule E), one could replace It(ϕ) by the corresponding Fitting ideal of E and t0(ϕ) by

t0 = t0(E) := inf{t0(ϕ) | Rm ϕ→ Rn → E → 0}.If need, we will work from a fixed presentation Rm

ϕ→ Rn → E → 0 for which t0(ϕ) =t0(E).

Our first result shows that t0(E) is closely related to the dimension of Sym(E).

Proposition 8.1. Let R be a Cohen–Macaulay local ring, and let E be an R-module witha rank. Then the following are equivalent:

(a) dim Sym(E) = dimR+ rk(E).

(b) t0(E) = 1.

(c) ν(EP ) ≤ ht(P ) + rk(E) for every prime P of R.

Proof. The equivalence of (a) and (b) is proved in [44, 2.1, 2.3]. The equivalence of (b)and (c) is a consequence of the more general result.

Lemma 8.2. Let R be a Cohen–Macaulay ring and let E be an R-module with a rank anda given presentation Rm

ϕ→ Rn → E → 0. Then, for any given integer k ≥ 0, the followingconditions are equivalent:

(a) gradeIt(ϕ) ≥ rk(ϕ)− t+ k + 1, 1 ≤ t ≤ rk(ϕ).

(b) ν(EP ) ≤ ht(P ) + rk(E)− k, for every prime ideal P such that EP is not RP -free.

Proof. First, for any P, we bring the original presentation to the form:

R`P ⊗RrPid⊗ϕ′→ R`P ⊗RνP → EP → 0, ν = ν(EP ), ` = n− ν.

The matrix ϕ′ has entries in PRP (hence r = 0 ⇔ EP is RP -free). From this one clearlyhas: It(ϕ) ⊂ P ⇔ t ≥ `+ 1.

(a)⇒(b): Let P be such that EP is not RP -free. Then ν(EP ) > rk(E), so ` + 1 =n− ν(EP ) + 1 ≤ n− rk(E) = rk(ϕ) and I`+1(ϕ) ⊂ P. We then have from the assumption:

ht(P ) = gradeI`+1(ϕ) ≥ rk(ϕ)− (`+ 1) + (k + 1) = n− rk(E)− `+ k =

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(n− `)− rk(E) + k = ν(EP )− rk(E) + k

(b)⇒(a): Given 1 ≤ t ≤ rk(ϕ), pick P ⊃ It(ϕ) such that ht(P ) = ht(It(ϕ)). In particularIrk(ϕ)(ϕ) ⊂ It(ϕ) and hence EP is not RP -free. Therefore, by assumption and the aboveremark:

gradeIt(ϕ) = ht(P ) ≥ ν(EP )− rk(E) + k = n− `− rk(E) + k ≥

n− (t− 1)− rk(E) + k = (n− rk(E))− t+ k + 1.

Remark 8.3. In the above lemma, condition (b) may be replaced by:

(b’) ν(Ep) ≤ ht(P ) + rk(E)− k, for every prime P such that ht(P ) ≥ k, provided E isgiven to be free in codimension ≤ k−1. Since E is always assumed to be free in codimension0, (b)⇔(b’) for k = 1. We will apply this in the following corollary.

Corollary 8.4. Let R be a Cohen–Macaulay ring and let I be an ideal of height > 0.Assume:

(i) I is generically a complete intersection.

(ii) R/I is Cohen–Macaulay.

Then the following are equivalent:

(a) gradeIt(ϕ) ≥ (n − 1) − t + 3, 1 ≤ t ≤ n − ht(I). (Note Rmϕ→ Rn → I → 0 is a

presentation of I.)

(b) ν(IP ) ≤ ht(P )− 1, for every prime P ⊂ I, ht(P ) ≥ ht(I) + 1.

Proof. Note that by assumption the R/I-module I/I2 has a rank (= ht(I)).

It suffices to apply 8.2 along with the above remark, provided w show that:

1) Condition (a) is equivalent to

grade(It(ϕ∗)) ≥ rk(ϕ∗)− t+ 2, 1 ≤ t ≤ rk(ϕ∗),

where

(R/I)mϕ∗→ (R/I)n → I/I2 → 0.

For this note that I ⊂√In−1(ϕ) ⊂

√It(ϕ), 1 ≤ t ≤ n − 1 and that rk(ϕ∗) = n −

rk(I/I2) = n− ht(I). Since R/I is Cohen–Macaulay, we get

gradeIt(ϕ) = grade(It(ϕ∗)) + ht(I), 1 ≤ t ≤ n− 1.

2) Condition (b) is equivalent to:

ν((I/I2)P ) ≤ ht(P/I) + rk(I/I2)− 1

for every prime P ⊃ I with ht(P ) ≥ ht(I) + 1.

But the equalities ν(IP ) = ν((I/I2)P ) and ht(P/I) = ht(P )−ht(I) translate (2) in (b).

The conditions of 8.1 are realized when Sym(E) is a Cohen-Macaulay ring. More gen-erally:

31

Proposition 8.5. Let R be a Cohen-Macaulay ring of finite Krull dimension and let E bean R-module with a rank. Then:

(i) If Sym(E) is unmixed (meaning: (0) is height unmixed ), dim Sym(E) = dimR +rk(E).

(ii) The following conditions are equivalent:

(a) Sym(E) is R-torsion-free.

(b) Sym(E) is unmixed and ν(EP ) ≤ ht(P ) + rk(E)− 1 for every prime P such thatht(P ) ≥ 1.

The proof of this result is essentially given in [44, 2.4 and 3.3] once the results there aretranslated by means of 8.2.

The main application of the above proposition will be given in the next section. For themoment, we single out some simple consequences.

Corollary 8.6. Let R be a Cohen–Macaulay ring and let I be an ideal of height > 0.

(a) If Sym(I) is Cohen–Macaulay, then ν(IP ) ≤ ht(P ) + 1 for every prime P.

(b) If I is generically a complete intersection and R/I and Sym(I/I2) are Cohen–Macaulayrings, then ν(IP ) ≤ ht(P ) for every prime P ⊃ I.

9 Cohen–Macaulay blowing-up rings

One of the major applications of the approximation complexes is to certain arithmeticalaspects of the blowing-up rings. More exactly, the acyclicity of these complexes, along witha stronger depth requirement on the Koszul homology modules, implies that all blowing-uprings are Cohen–Macaulay. In a sense, the Cohen–Macaulayness of the algebra Sym(I)alone is to trigger the same property for the other blowing-up rings. This phenomenontransfers the burden of checking arithmetical properties to one single algebra; it is howeverunder Cohen-Macaulayness of the Koszul homology modules that one meets most ‘block’examples (cf. §10).

We first give a version that uses the approximation complex M.

As before, ν(E) denotes the minimal number of generators of the module E and ht(I)stands for the height of the ideal I. Hi is the i-th Koszul homology module relative to a setof generators of I.

Theorem 9.1. Let R be a Cohen–Macaulay ring and let I be an ideal of height > 0. Assume:

(i) ν(IP ) ≤ ht(P ) for every prime P ⊃ I.

(ii) depth(Hi)P ≥ ht(P ) − ν(IP ) + i, for every prime P ⊃ I and very 0 ≤ i ≤ ν(IP ) −ht(IP ).

Then:

(a) Sym(I) ' R(I) and Sym(I/I2) ' grI(R).

32

(b) Sym(I) and Sym(I/I2) are Cohen–Macaulay rings.

(c) If it is further the case that R is Gorenstein and the Hi are Cohen–Macaulay, thenSym(I/I2) is a Gorenstein ring.

Note that (ii) is in particular satisfied if the Hi are Cohen–Macaulay.Proof. (i) and (ii) imply that depth(Hi)P ≥ i for 0 ≤ i ≤ ν(IP )− ht(IP ). Since outside ofthis range (Hi)P is certainly free, we see that condition (b) of 5.1 is satisfy. Therefore thecomplex M is acyclic and this warrants (a) (cf. §4).

To show (b), it suffices to handle the case where R is local. Say, dimR = d. Thendim Sym(I) = dimR(I) = d + 1 and dim Sym(I/I2) = dim grI(R) = d. We have thereforeto show that depth Sym(I) = d+ 1 and depth Sym(I/I2) = d.

To see this we apply 5.2 (b) to the complexes Z and M, respectively (N.B. Since Mis acyclic, so is Z (cf.§4), but this could be checked directly using 5.4). Namely the depthconditions on the Hi imply the following depth estimates

depth(Hi ⊗ (R/I)[T1, . . . , Tn]) ≥ d+ i, 0 ≤ i ≤ s

depth(Zi ⊗R[T1, . . . , Tn]) ≥ d+ 1 + i, 0 ≤ i ≤ n− 1

where n = ν(I), s = ν(I) − ht(I) and depths are taken with respect to the irrelevantmaximal ideal of the polynomial ring R[T1, . . . , Tn]. This is exactly what is needed to apply5.2(b)

Finally, (c) follows from the self-duality in this context of M. Namely, we must showthat ExtsS(Sym(I/I2),KS) = Sym(I/I2), where s = ν(I) − ht(I), S = (R/I)[T1, . . . , Tn]and KS = HS ⊗ S is the canonical module of S. By [15] this is exactly the condition forthe Cohen–Macaulay ring Sym(I/I2) to be Gorenstein. The self-duality of M in turn is aconsequence of the fact that, with Hi Cohen–Macaulay, the homology algebra H = H•(K)is a Poincare algebra (cf. [14]).

Remark 9.2. A major source of applications of the preceding theorem is provided by thetheory of linkage (cf. §7), but there are other instances as well (cf. §5).

We next give a version that does not involve homology at all.

Proposition 9.3. Let R be a Cohen–Macaulay ring and let I be an ideal of height > 0.Assume:

(i) Sym(I) and R/I are Cohen–Macaulay rings.

(ii) I is generically a complete intersection.

The following conditions are equivalent:

(a) Sym(I) ' R(I).

(b) Sym(I/I2) is a Cohen–Macaulay ring.

(c) ν(IP ) ≤ ht(P ) for every prime P ⊃ I.

Moreover, the following conditions are equivalent:

33

(α) Sym(I/I2) is R/I-torsion free.

(β) ν(IP ) ≤ ht(P ) for every prime P ⊃ I such that ht(P ) ≥ ht(I) + 1.

(γ) htFs(I) ≥ s+ 1, ht(I) + 1 ≤ s ≤ ν(I) (F·(I) = Fitting ideals of I)

Remark 9.4. In the latter set of equivalent conditions, if in addition I is a prime idealthen ordinary and symbolic powers coincide. Such applications were the ‘leitmotiv’ forinvestigating the blowing-up rings ([43]).

Proof. (sketch) Most of the ground has been prepared in the preceding section. To get(b)⇒(c) apply the equality dim Sym(I/I2) = dim(R/I) + rk(I/I2) obtained by 8.5. To get(c)⇒(a) apply the implication (b)⇒(a) of 8.5 with E = I/I2 and observe that the kernelSym(I) → R(I) is annihilated by a power of I, an ideal containing a nonzero divisor byassumption. The equivalence (α) ⇔ (β) is also a straightforward application of 8.5, whilethe equivalence (β)⇔ (γ) is a reformulation of 8.4. Finally, the only implication that needsa different argument is (a)⇒(b). For this we assume R local and observe that, because ofthe isomorphisms Sym(I) ' R(I) and Sym(I/I2) ' grI(R), we may compute the depth ofSym(I/I2) from the exact sequences

0→ IR(I)→ R(I)→ grI(R)→ 0

and

0→ R(I)+ → R(I)→ R→ 0,

with repect to the irrelevant ideal of R(I). Clearly, IR(I) ' R(I)+, so we get depthgrI(R) = depth R, as required.

Remark 9.5. The above proof actually shows that, without the assumption that either R/Ibe Cohen–Macaulay or that I be generically a complete intersection, the conditions “Sym(I)is Cohen–Macaulay”and “ν(IP ) ≤ ht(P ) for every prime P ⊃ I”imply Sym(I) ' R(I) andSym(i/I2) Cohen–Macaulay. Thus, the preceding theorem may be viewed as a naturalnon-homological substitute for 9.1.

In generic it is not true that Sym(I) is Cohen–Macaulay if Sym(I/I2) is Cohen–Macaulay,as easy example show. Even if we add that I be generically a complete intersection, theimplication fails to hold - an example will be given in the next section. We don’t knowhowever if this is the case if, moreover, R/I is Cohen–Macaulay.

10 Cohen–Macaulay symmetric algebras

In the preceding section we pointed to the interest of knowing when Sym(I) is a Cohen–Macaulay ring (cf. 9.3). In the present one, we will make a few variations on this theme.We start with a criterion of acyclicity of the complex Z.

Theorem 10.1. Let R be a Cohen–Macaulay ring and let I be an ideal of height > 0.Assume:

(a) ν(IP ) ≤ ht(P ) + 1, for every prime ideal P.

34

(b) depth(Hi)P ≥ ht(P ) − ν(IP ) + i, for every prime ideal P and every i such that0 ≤ i ≤ ν(IP )− ht(Ip).

Then Sym(I) is Cohen–Macaulay.

Proof. Consider the complex Z associated to I. By 5.4 we know that Z is acyclic. Depth-chasing in a way entirely similar to that in the proof of 9.1, yields depth Sym(I) ≥ dimR+1.On the other hand, by 8.1, dim Sym(I) ≤ dimR+1. Therefore, Sym(I) is Cohen–Macaulay.

Note that the above theorem is analogous to 9.1, except that we stay away from theequality Sym(I) = R(I).

We next give an applications which extends some results of Rossi ([37]).

Corollary 10.2. Let (R,m) be a Cohen–Macaulay local ring of dimension > 0 and let I bean m-primary dieal. The following are equivalent:

(a) I is an almost complete intersection.

(b) Sym(I) is Cohen–Macaulay.

Proof. (a)⇒(b) follows from 10.1, while the reverse implication issues from 8.6.Next is a variant of 10.1, in terms of sequential conditions.

Proposition 10.3. Let R be a Cohen–Macaulay local ring and let I be an ideal of height> 0. Assume:

(i) ν(IP ) ≤ ht(P ) + 1 for every prime P.

(ii) I is generated by a proper sequence {x1, . . . , xn} satisfying depth(R/(x1, . . . , xj)) ≥depth R− j, 1 ≤ j ≤ n and depth R/I ≥ dimR/I − 1.

Then Sym(I) is Cohen–Macaulay.

Proof. Since I = (x1, . . . , xn) and the xi form a proper sequence, the Z-complex is acyclic(cf. §12). The present depth estimates imply, moreover, that

depth Hj(x;R) ≥ dimR+ j − n, for 1 ≤ j ≤ n,

(cf. 7.1).Now, depth-chasing as in the proof of 9.1, we get depth(Zi ⊗ R[T1, . . . , Tn]) ≥ dimR +

i+1. By 5.2 (b) depth Sym(I) ≥ dimR+1. Finally, as before, (i) means that dim Sym(I) =dimR+ 1.

As a consequence, we can derive a handy criterion in the case I is an almost completeintersection.

Corollary 10.4. Let R be a Cohen–Macaulay local ring and let I be an ideal of height > 0.Assume:

(a) I is an almost complete intersection.

(b) depth R/I ≥ dimR/I − 1.

35

Then Sym(I) is Cohen–Macaulay.

Proof. By (a) I is generated by {x1, . . . , xn−1, xn} where the first n − 1 elements forma regular sequence; this is certainly a proper sequence. Furthermore, (b) provides for theonly depth requirement needed. Finally, an almost complete intersection always satisfiesthe requirement ν(IP ) ≤ ht(P ) + 1, for any prime P.

Remark 10.5. In 10.4 I is not assumed to be generically a complete intersection. If thatis further granted and R is moreover Gorenstein, then Sym(I/I2) is Cohen–Macaulay aswell. In fact, in the presence of the condition depth R/I ≥ dimR/I−1, it must then be thecase that depth R/I = dimR/I, so R/I is Cohen–Macaulay (we thank C. Huneke for thisobservation). But, if this is case we can apply 9.3 to get Sym(I) ' R(I) and Sym(I/I2)Cohen–Macaulay. It may be worthwhile to add that if I is a complete intersection only atits primes of minimal height, then R/I is not necessarily Cohen–Macaulay yet Sym(I/I2)is Cohen–Macaulay and Sym(I) ' R(I) (cf. [44, §5] for both an example and the proof ofthis fact).

Remark 10.6. Criterion (i) of 10.3 is not superfluous in the presence of a generating setthat is a proper sequence and satisfies the depth requirements stated there. For example,take I to be the maximal ideal of a local ring, whose square is zero. However, it maybe the case that (i) is superfluous under the assumption that I be generically a completeintersection.

In the sequel we show, in the vein of [9, 1.1], under what conditions the Cohen–Macaulayproperty of the symmetric algebra is preserved under specialization.

Precisely, let R be a Cohen–Macaulay ing and let I be an ideal of height > 0. Weassume an ideal N is given such that: (i) N is perfect, and (ii) I is not contained in anyassociated prime of N. We then ask when Sym(I) Cohen–Macaulay implies Sym(I/NI)Cohen–Macaulay. Note that (ii) implies that I/NI is (an R/N -module) with generic rankone. In the applications N will actually be generated by an R/I sequence so that I/NI 'I +N/N ⊂ R/n in this case.

Proposition 10.7. Let I and N be given as above. If Sym(I) is Cohen–Macaulay, thenthe following are equivalent:

(a) Sym(I/NI) is Cohen–Macaulay.

(b) ν(IP ) ≤ ht(P/N) + 1, for every prime P ⊃ I +N.

Proof. (a)⇒(b): Since ν(IP ) = ν(I/NI)P holds, this implication follows directly from 8.1and 8.5.

(b)⇒(a): By the usual argument (cf. [34]) it suffices to show that ht(N · Sym(I)) ≥ht(N). To arrive at this, we first appeal to 8.5 as applied to Sym(I), to get dim Sym(I) =dimR+1. Then we apply 8.1 to Sym(I/NI) under condition (b). Putting the pieces togetherwe find ht(N · Sym(I)) = ht(N).

Remark 10.8. Note that the conditions on N imply TorR1 (I,R/N) = 0 which, however,may not suffice to make I/NI isomorphic to an ideal of R/I. For this we need the strongerrequirement ν(IP ) ≤ ht(P/N) as in 8.5 (ii).

36

The following may be viewed as an application of 10.7 - but follows also from 8.6, 10.1and the results of §7. It is analogous to [9, 3.4], but note the present hypotheses do notnecessarily imply the equality of the symmetric and Rees algebra.

Corollary 10.9. Let R be an Cohen–Macaulay ring and let I be an ideal of finite projectivedimension. Assume either:

(i) ht(I) = 2 and R/I is Cohen–Macaulay, or

(ii) ht(I) = 3 and R/I (and R) are Gorenstein.

Then the following conditions are equivalents:

(a) Sym(I) is Cohen–Macaulay.

(b) ν(IP ) ≤ ht(P ) + 1, for every prime P.

Example 10.10. Consider the matrix

ϕ =

x y z 00 x y z0 0 x y

(where R = k[[x, y, z]]).

Here, I = I3(ϕ) = (x3, x2y, x(y2 − xz), y3), ht(I3(ϕ)) = 2, ht(I2(ϕ)) = 3, ht(I1(ϕ)) = 3.Therefore, by 8.1 and the preceding corollary, we get that Sym(I) is Cohen–Macaulay.(Note, however, that Sym(I) 6= R(I) as ν(I) = 4 > dimR.)

An similar example of case (ii) can be gotten with a 5 × 5 symmetric matrix in apolynomial ring with 4 variables. It is quite conceivable that for any given n ≥ 3 there existexamples of n × (n + 1) matrices (resp. 2n − 1 × 2n − 1 skew-symmetric matrices) whoseideal of n × n minors (resp. 2n − 2 × 2n − 2 pfaffians) has a Cohen–Macaulay symmetricalgebra and, moreover, not coinciding with the corresponding Rees algebra.

We now wish to point out that, contrarily to what happens in the favourable habitat ofRess algebras, symmetric algebras tend to be non-Cohen–Macaulay in some very standardgeneric situations.

(1) Determinantal ideals: Let (xij), 1 ≤ i ≤ m, 1 ≤ j ≤ n, be a matrix of indeterminatesover a field. Assume m ≥ n + 2. Let I be the ideal of n × n minors. Then, we know thatSym(I) 6= R(I). Moreover, if

(mn

)≥ mn + 2 then we know that Sym(I) is not Cohen–

Macaulay because this blasts the number of generators condition. Say, n = 2. In this case,it is easy to see that I is a complete intersection in the punctured spectrum of R(= k[[xi,j ]]).It follows that depth Hi = 0, so that we are far from being able to apply any of the so-fargiven criteria for the acyclicity of eitherM or Z. In fact, Sym(I) is never Cohen–Macaulayin this case: If m ≥ 6 then ν(I) = dimR+ 3; if m ≤ 5, then ν(IP ) ≤ ht(P ) for every P ⊃ I,so that Sym(I) is again not Cohen–Macaulay as otherwise it would be torsion-free by 8.5.

(2) Divisors scrolls. (cf. [9]) These are given as follows: Pick a generic 2× 2 matrix

X =

[x11 x12x21 x22

]37

and a ‘semi-generic’ 2× 2 matrix

Y =

[y11 y12y21 y22

]and consider the 2× 4 matrix (X,Y X). Let I be the ideal generated bib the 2× 2 minorsof this matrix. We claim that Sym(I) is not Cohen–Macaulay (not even unmixed).

To see this, we first observe that the estimate ν(IP ) ≤ ht(P ) holds for every primeP ⊃ I. Indeed, this is obvious if ht(P ) ≥ 4. If ht(P ) = 2, then P = I so that IP is acomplete intersection in RP . Finally, if ht(P ) = 3 then (x11, x12, x21, x22) 6⊂ P ; say x11 /∈ P.By effecting an elementary transformation in the matrix (X,YX), we set it in the form[

1 0 0 00 #1 #2 #3

]form which it is clear that ν(IP ) ≤ 3.

On the other hand, Sym(I) is not torsion–free (in fact, the Plucker relation amongthe generators of I shows that S2(I) 6= I2). Therefore, by 8.5 (ii), Sym(I) is not Cohen–Macaulay.

It follows from 10.7 (checking (b)) that if I is the homogeneous ideal of the smoothrational non-normal quartic in P3, then Sym(I) is not Cohen–Macaulay (of course, thiscould also be checked directly as above).

As for Sym(I/I2), the situations are different in (1) as in (2). We have:(1)’ Let I ⊂ R = k[[xij ]], 1 ≤ i ≤ 2, 1 ≤ j ≤ 4, be the ideal of the 2 × 2 minors of thegeneric 2× 4 matrix over the field. Then Sym(I/I2) is not Cohen–Macaulay.Proof. We have ν(I) = 2 < 7 = dimR − 1. Also, as remarked in (1), I is a completeintersection of height three in the punctured spectrum of R. Therefore, for a prime P suchthat 4 ≤ ht(P ) ≤ 7, we have ν(IP ) = ht(IP ) = ht(I) ≤ ht(P ) − 1. Since R/I is Cohen–Macaulay, if Sym(I/I2) were Cohen–Macaulay we could have, by 8.5(ii), that Sym(I/I2)is domain. Clearly, we have dim Sym(I/I2) = dimR. On the other hand, we know thatSym(I/I2) 6= grI(R). It follows that dim Sym(I/I2) ≥ dim grI(R) + 1 = dimR + 1, acontradiction.(2)’ Let I ⊂ R = k[[x, y, z, w]] be the ideal of the smooth rational non–normal quartic inP3. Then Sym(I/I2) is Cohen–Macaulay.Proof. We use the M∗–complex on a generating set of 4 elements:

0→ H∗2 ⊗ S → H∗1 ⊗ S → H∗0 ⊗ S → Sym(I/I2)→ 0.

Note that H∗2 = H2 = canonical module of R/I, and thus has depth 2 (cf. [15]). H∗1 ,on the other hand, is torsion–free, and thus has depth≥ 1. As the complex satisfies theconditions of 5.2, as in 5.1 we conclude that it is exact. Counting depths, we have depthSym(I/I2) ≥ 4 = dim Sym(I/I2), as asserted.

Remark 10.11. (2) gives an example of a generically complete intersection ideal I forwhich Sym(I/I2) is Cohen–Macaulay but Sym(I) is not. Note, however, that R/I is notCohen–Macaulay. Does there exist a similar example with R/I Cohen–Macaulay?

To conclude, a word on the property of being Gorenstein for the symmetric algebra.It was shown in §9 that if I ⊂ R is generically a complete intersection in a Gorenstein

38

ring and if one is given appropriate conditions that make the complex M acyclic, thenSym(I/I2) is a Gorenstein ring. The situation is dramatically diverse for Sym(I). here iteasy to see, by localization at a minimal prime of height = ht(I), that Sym(I) Gorensteinimplies ht(I) ≤ 2.

It is conceivable that this bound on the height of I works in general, even if I is notgenerically a complete intersection. This is the case if m is the maximal ideal of a hypersur-face (communicated by E. Rossi) and, conversely, we can use the acyclicity of Z, togetherwith the high depths of its components in this case, to conclude that if ht(m) = 2 thenSym(m) is Gorenstein (this has also been shown by Rossi).

11 The syzygetic reduction groups

In sections 2 and 6 we studied the δi(I)’s and some of their various interpretations. Thissection is devoted to extending such notions to obtain a whole array of measures of com-parison.

Here and in later sections it will be necessary to con sider the approximation complexeswith coefficients in a module (cf. §4).

Let I be an ideal of the Noetherian ring R and let M be a finitely generated R-module.On the Koszul complex K(I;M) -built on some generating set of I - one can define a naturalfiltration F∗ :

FjKi(I;M) =

{Ij−1Ki(I;M), for j ≥ 1Ki(I;M), for j ≤ i

Definition 11.1. The module δj1(I;M) = Hi(Fi+jK(I;M)) are called the syzygeticgroups of I with respect to M.

Note that δji (I;M) = Hi(I;M) for j ≤ i, and δ11(I;R) = δ1(I) in the terminology §2.

Of course, the δji (I;M) may depend on the choice of the generators of I, since this iscase of K(I;M) itself. However, we will see below that a certain invariance property holds.

There are two other obvious ways to describe the δji (I;M). For simplicity, let K =K(I,M), Z = Z(I;M) and B = B(I;M).

Lemma 11.2. For j ≥ 1, we have

(a) δji (I;M) = Zi ∩ IjKi/Ij−1Bi.

(b) δji (I;M) = im(Hi(I; IjM)→ Hi(I; Ij−1M))

Lemma 11.3. If δji (I;M) = 0 for some i and j, then δji+1(I;M) does not depend on thechosen generators of I.

Proof. By standard arguments we may restrict ourselves to show the following: If x ={x1, . . . , xn} generates I, then the canonical mapping

ϕ : Hi+1(Fi+j+1K(x;M))→ Hi+1(Fi+j+1K(x, 0;M))

is an isomorphism.

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In K1(x, 0;M) we have a basis {e1, . . . , en, en+1} with ∂ei = xi for i = 1, . . . , n and∂en+1 = 0. We first show that ϕ is injective: In fact, let z ∈ Fi+j+1K(x;M) be a cycle andsuppose that z = ∂(b0 + b1 ∧ en+1) with b0, b1 ∈ Fi+j+1K(x;M). Then necessarily ∂b1 = 0and z = ∂b0.

To see that ϕ is surjective, let z = z0 + z1∧en+1 be an (i+ 1)-cycle of Fi+j+1K(x, 0;M)with z0 ∈ Fi+j+1Ki+1(x;M), and z1 ∈ Fi+jKi(x;M). it then follows that z1 is also a cycle

and hence a boundary, since we assume that δji (I;M) = 0. Pick b ∈ Fi+jKi+1(x;M) suchthat z1 = ∂b. Then we have b ∧ en+1 ∈ Fi+j+1Ki+2(x, 0;M) and ∂(b ∧ en+1) = z1 ∧ en+1.Hence z is homologous to z0 which lies in the image of ϕ.

Definition 11.4. The ideal I is called j-syzygetic with respect to M, if δki (I;M) = 0 forall i and all k with 1 ≤ k ≤ j.

By the previous lemma it is clear that this notion depends only on the ideal and not onthe chosen generators. It was first introduced by Huneke ([23]). The most interesting casewill be when I is j-syzygetic for all j.

Remark 11.5. (a) For any ideal I and any finitely generated module M there exists aninteger n0 such that I is j-syzygetic with respect to InM for all j and all n ≥ n0. Thisfollows from a simple application of the Artin–Rees lemma to the complexes FkK(I;M) (cf.[1]).

(b) It is clear from the definition that if I is j-syzygetic with respect to M for all j, thenso is I with respect to IM.

Let us give another interpretation of the syzygetic reduction groups: Having chosen aset of generators {x1, . . . , xn} of I we obtain a natural R-algebra presentation

S = R[e1, . . . , en]→ R(I), ei → xit = x∗i

of the Rees algebra of I. We fix this presentation. the irrelevant ideal S+ = (e1, . . . , en) ofS acts in a natural way on the R(I)-module R(I;M) =

⊕j I

jM and the grI(R)-module

grI(M) =⊕IjM/Ij+1M, which corresponds to the actions of the irrelevant ideal of R(I),

resp. grI(R).

Lemma 11.6. (a) Hi(S+;R(I;M))i+j

{δji (I;M), for j > 0Zi(I;M), for j = 0.

(b) There exists a long exact sequence

. . .→ δj+1i (I;M)→ δji (I;M)→ Hi(S+; grI(M))i+j → δj+1

i−1 (I;M)→ . . .

Proof. (a) is immediate.(b) We have

K(S+; grI(M)) ' grK(I;M) =⊕FjK(I;M)/Fj+1K(I;M)

one now considers the long exact homology sequence corresponding to the isomorphism.Some of these syzygetic groups may also be interpreted in terms of the syzygy part of

the Koszul homology (cf. [42]; see also §6).In our terminology H∗i (I;M) ' Coker(δ1i (I;M)→ δ0i (I;M)). Since the mapping above

is always injective, we obtain

40

Corollary 11.7. H∗i (I;M) ' Hi(S+; grI(M))i

Next we show that the syzygetic reduction groups measure the homology of the approx-imation complex M(I;M). It is convenient to consider M(I;M) with its natural augmen-tation mapping

∂′0 :M(I;M)→ grI(M)

where ∂′0 is the composition

H0(I;M)⊗ S → H0(M(I;M))β→ grI(M).

Thus we obtain an extended complex

N : . . .→M1(I;M)→M0(I;M)→ grI(M)→ 0.

It is then clear that H−1(N ) = 0 if and only if β is an isomorphism - that is, if I is of lineartype with respect to M (4.10).

Our aim is to express the first non-vanishing homology of N in terms of the δji .

Theorem 11.8. Suppose that for some i ≥ 0 we have H`(N ) = 0 for all ` < i, then:

(a) H`(S+; grI(M)) ' H`(I;M)(`), for ` ≤ i.

(b) (Hi(N )/S+Hi(N ))j =

{0, for j < i and j = i+ 1

Hi+1(S+; grI(M))j , for j > i+ 1

Note that, from the definition, one has

(Hi(N )/S+Hi(N ))i = Hi(N )i '

{0, for i = 0

ker (Hi(I;M)∂′→ Hi−1(I;M)⊗ S1), for i > 0.

Before proving the theorem we draw some consequences. First we reprove a result ofM. Kuhl ([27, 2.7])

Corollary 11.9. I is an ideal of linear type with respect to M, if only if δj1(I;M) = 0 forj ≥ 1.

By the theoremH0(N ) = 0 inf and only if δ11(I;M) = ker (H1(I;M)→ H0(I;M)⊗S1) =0 and H1(S+; grI(M))j = 0 for j > 1. By 11.6 we have the exact sequence

δj+11 (I;M)→ δj1(I;M)→ H1(S+; grI(M))i+j → 0.

But, as remarked earlier, δk1 (I;M) = 0 for k >> 0, by the Artin–Rees lemma, and thus theassertion.

Corollary 11.10. The following conditions are equivalent:

(a) M(I;M) is acyclic.

(b) I is j-syzygetic with respect to M for all j ≥ 1.

41

(c) Hi(S+; grI(M)) ' Hi(I;M)(i) for all i ≥ 0.

Proof. The equivalence of (b) and (c) follows immediately from 11.6.

In §4 we have seen that I is of linear type with respect to M if M(I;M) is acyclic.Hence M is acyclic if and only if N is exact. Now the equivalence of (a) and (c) followseasily from 11.8.

Proof. (of 11.8) by assumption we have the exact sequence

0→ Ui → Ni → . . .→ N0 → N−1 → 0.

Since H`(S+;Nk) = 0 for k ≥ 0 and ` > 0, this exact sequence gives rise to isomorphismsof graded modules

(1) H`(S+; grI(M)) ' H0(S+;N`) for ` ≤ i,

(2) Hi+1(S+; grI(M)) ' ker (H0(S+;Ui)→ H0(S+;Ni)).

Since H0(S+;N`) ' H`(I;M)(`), the assertion (a) of the theorem follows from (1). (2), onthe other hand, implies

(3) Hi+1(S+; grI(M))j ' H0(S+;Ui) for j > 1.

For all j we also have the exact sequence

(4) H0(S+;Ni+1)j → H0(S+;Ui)j → H0(S+;Hi(N ))j → 0

Again, since H0(S+;Ni+1)j = 0 for j > i + 1, we obtain from (4) H0(S+;Ui) 'H0(S+;Hi(N ))j for j > i + 1, which in conjunction with (3) proves part of (b). It isclear that H0(S+;Hi(N ))j = 0 for j < i, since Hi(N ) is a sub quotient of Ni. It remainstherefore to show that H0(S+;Hi(N ))i+1 = 0. This follows from the commutative diagramwith exacts rows

H0(S+;Ni+1)i+1// H0(S+;Ui)i+1

// H0(S+;Hi(N ))i+1// 0

δ0i+1(I;M) // Hi+1(S+; grI(M))i+1// δ1i (I;M) // δ0i (I;M)

In fact, since δi(I;M) → δ0i (I;M) is trivially injective, it follows that H0(S+;Ni+1)i+1

is surjective, and H0(S+;Hi(N ))i+1 = 0, as asserted.

12 Sequences and the approximation complexes

In §3 we have seen that ideals which are generated by d-sequence are ideal of linear type.In this section we want to show much more. If (R,m, k) is a Noetherian local ring whoseresidue field k is infinite, then for an ideal I and a finitely generated R-module M, we showthat the following are equivalent:

(a) M(I;M) is acyclic.

(b) I is generated by a d-sequence with respect to M.

42

The implication (b)⇒(a) has been shown already in [18] -without the restriction on k. Also,under some additional Cohen–Macaulay conditions, Huneke ([23]) proved the equivalenceof (a) and (b).

The notion of a d-sequence that we shall use is 6.1(a∗2) as it is convenient to skip theminimality condition.

For several reasons, we need some other characterizations of d-sequences.

Lemma 12.1. Let x = {x1, . . . , xn} be a sequence of elements generating the ideal I. Thefollowing conditions are equivalent:

(a) x is a d-sequence with respect to M.

(b) (x1, . . . , xi)M : xi+1 ∩ IM = (x1, . . . , xi)M, for i = 1, . . . , n− 1

(c) (x1, . . . , xi)IM : xi+1 ∩ IM = (x1, . . . , xi)M and (x1, . . . , xi)IM : xi+1 ∩ I2M =(x1, . . . , xi)IM, for i = 1, . . . , n− 1.

We have picked only a few of the possible characterizations of d-sequences which arelisted in [27, 1.2], The implication (a)⇒(b) is due to Huneke ([20]). The equivalence of (a)and (c) was first observed by Kuhl ([27]). Note that a sequence satisfying the first equationof (c) is a relative regular sequence with respect to M (cf. 6.1). The second equation of (c)says that x is a d-sequence with respect to IM. Thus we may rephrase 12.1 as follows.

Corollary 12.2. The following conditions are equivalent:

(a) x is a d-sequence with respect to M.

(b) x is a relative regular sequence with respect to M and a d-sequence with respect toIM.

(c) For all k ≥ 0, x is a relative regular sequence with respect to IkM and a d-sequencewith respect to Ik+1M.

In fact, (c) fullows by induction from (a) and (b).Before proving 12.1, let us draw one interesting consequence of 12.2. We will use a result

of Fiorentini ([11]).

Theorem 12.3. If I is generated by a relative regular sequence with respect to M, then Iis 1-syzygetic with respect to M.

From 12.2 it now follows immediately

Corollary 12.4. If I is generated by a d-sequence with respect to M, then I is j-syzygeticwith respect to M for all j, and hence M(I;M) is acyclic (by 11.10).

This is a new proof of ([18, 5.6]) and also a new proof of the fact (cf. §3) that if I isgenerated by a d-sequence with respect to M, then I is of linear type with respect to M.Proof. (of 12.1) (a)⇒(b): If a ∈ (x1, . . . , xi)M : xi+1 ∩ IM then aIM ⊂ (x1, . . . , xi)M.In fact, if k ≥ i + 1, then axi+1xk ∈ (x1, . . . , xi)M and hence axk ∈ (x1, . . . , xi)M, i.e.axk ∈ (x1, . . . , xi)M for all k. Now suppose a =

∑kj=1 ajxj , ai ∈M. If k ≥ i+1, then axk ∈

(x1, . . . , xi)M ⊂ (x1, . . . , xk−1)M, and hence akx2k ∈ (x1, . . . , xk−1)M. But this implies that

43

akxk ∈ (x1, . . . , xk−1)M. Thus we can rewrite a =∑k−1

j=1 a′jxj . By inverse induction on k we

conclude that a ∈ (x1, . . . , xi)M. The other inclusion is trivial.

(b)⇒(c): The first equality of (c) follows trivially from (b). To prove the second one weshow that

(x1, . . . , xi)M ∩ I2M = (x1, . . . , xi)IM for i = 1, . . . , n (2)

From (2) the second equality of (c) easily. In fact,

(x1, . . . , xi)IM : xi+1 ∩ I2M = ((x1, . . . , xi)IM : xi+1 ∩ IM) ∩ I2M = (x1, . . . , xi)M ∩ I2M = (x1, . . . , xi)IM.

(b)⇒(2): Let a ∈ (x1, . . . , xi)M ∩ I2M, a =∑k

j=1, aj ∈ IM. Suppose that k ≥ i + 1;then akxk ∈ (x1, . . . , xk−1)M and hence ak ∈ (x1, . . . , xk−1)M : ak∩IM = (x1, . . . , xk−1)M.Thus a =

∑k−1j=1 a

′jxj . By induction it follows that (x1, . . . , xi)M ∩ I2M ⊂ (x1, . . . , xi)IM.

Again, the other inclusion is trivial.

(c)⇒(a): We are going to show that for all k ≥ i+ 1 we have (x1, . . . , xi)M : xi+1xk ⊂(x1, . . . , xk−1)M : xk. This implies easily that x is a d-sequence.

Let a ∈ (x1, . . . , xi)M : xi+1xk; then axi+1xk ∈ (x1, . . . , xi)M, and hence ax2k ∈(x1, . . . , xi)IM : xi+1 ∩ I2M = (x1, . . . , xi)IM = (x1, . . . , xk−1)IM . It follows that axk ∈(x1, . . . , xk−1)IM : xk ∩ IM = (x1, . . . , xk−1)IM

As it was pointed out in §6, if x is a relative regular sequence with respect to M, itfollows from Fiorentini’s theorem that x forms a proper sequence with respect to M.

Theorem 12.5. Let I be an ideal generated by a proper sequence with respect to M. ThenZ(I;M) is acyclic.

Proof. (cf. [18]) We let {e1, . . . , en} denote the standard basis of Rn, and the basis ofS1(R

n) as well (cf. §4). K and Z will stand of the Koszul complex K(x;M) and its moduleof cycles Z(x;M).

Let z ∈ Z; since we view the elements of Z as polynomials in e1, . . . , en with coefficientsin Z, we can write

z =∑

zh,i ⊗ heij

where the h’s are distincts monomials in e1, . . . , ej−1, and ∂(zh,i) = 0 for all h, i. We proveby induction on j that z = ∂′(b), with b ∈ Z.

j = 1 : We write simply

z =∑

zi ⊗ ei1.

We claim that zi ∈ Z(x1) for all i. Indeed, let k = inf{` | zi ∈ Z(x1, . . . , x`)}. Assume k > 1.Each zi can then be written as

zi = z′i + w′i ∧ ek, with z′i, w′i ∈ K(x1, . . . , xk−1).

Since

0 = ∂′(z) =∑

w′i ⊗ ei1ek + (terms in e1, . . . , ek−1),

it follows that w′i = 0. Thus z − i ∈ K(x1, . . . , xk−1), a contradiction.

44

We now havez =

∑aie1 ⊗ ei1, aix1 = 0.

Since 0 = ∂′(z) =∑aie

i+11 , it follows that z = 0.

j > 1 : As in the preceding case, it can be shown that zh,i ∈ Z(x1, . . . , xj). We nowprove by induction on degej (z) = t, that z = ∂′(b).

If t = 0, then by our induction hypothesis on j, we are done. Suppose then t > 0. Wewrite

zh,t = z′h,t + w′h,t ∧ ej

as before (if zh,t ∈ Z1(K), then zh,t = z′h,t + w′h,tej , w′h,t ∈ R). Then

0 = ∂′(z) =∑h

w′h,t ⊗ het+1j + (terms of lower ej − degree).

It follows that all w′h,t = 0 and hence zh,t ∈ Z(x1, . . . , xj−1).On the other hand, we have

∂(zh,t ∧ ej ⊗ het−1j ) = xjzh,t ⊗ het−1j .

Since x is proper, there exists bh,t ∈ K(x1, . . . , xj−1) with ∂(bh,t) = xjzh,t. Let

b =∑h

(zh,t ∧ ej ⊗ het−1j − bh,t ⊗ het−1j ).

Then b ∈ Z and degej (z − ∂′(b)) < t. By induction z − ∂′(b) is a boundary and the proof iscomplete.

As this proof indicates, the notion of a proper sequence is closer to the definition of theapproximation complexes than the other kinds of sequences. The next proposition showsrelated aspects. Let H+(x1, . . . , xi;M) =

⊕j>0Hj(x1, . . . , xi;M).

Proposition 12.6. Let x = {x1, . . . , xn} be a sequence generating the ideal I. The followingconditions are equivalent:

(a) x is a proper sequence with respect to M.

(b) IH+(x1, . . . , xi;M) = 0 for i = 1, . . . , n− 1.

(c) The canonical homomorphisms

H+(x1, . . . , xi;M)→ H+(x1, . . . , xi+1;M)

are injective for i = 1, . . . , n− 1.

It is quite surprising that 12.4 and 12.5 have a converse - for R local with infinite residuefield. In other words, theM-complex recognizes ideals which are generated by d-sequences,while the Z-complexes recognizes ideals which are generated by proper sequences. to provethis we first need two technical results.

Lemma 12.7. Let S = S0 ⊕ S1 ⊕ . . . be a positively graded S0-algebra generated by the 1-forms x1, . . . , xn. Let M be a graded S-module with (M/S+M)i = 0 for i 6= 0. The followingconditions are equivalent:

45

(a) {x1, . . . , xn} is a d-sequence with respect to M.

(b) {x1, . . . , xn} is a relative regular sequence with respect to M.

(c) Hj(x1, . . . , xn;M)k = 0 for j > 0, k 6= j and all i = 1, . . . , n.

(d) H1(x1, . . . , xn;M)k = 0 for k 6= 1 and all i = 1, . . . , n.

Proof. (a)⇒(b) has been remarked before.(b)⇒(c) (cf. [27, 2.3]): By Fiorentini’s theorem, x1, . . . , xi is 1-syzygetic for all i. Hence if

z is a homogeneous cycle of deg(z) > j representing an element of Hj(x1, . . . , xi;M), then itscoefficients lie in S+M and thus are all boundaries. This shows that Hj(x1, . . . , xi;M)k = 0for k > j. That Hj(x1, . . . , xi;M)k = 0 for k < j is trivial.

(d)⇒(a): Let a ∈ (x1, . . . , xi−1)M : xi ∩ S+M be a homogeneous element; Then axi =∑ajxj , 1 ≤ j ≤ i − 1, where the aj are homogeneous laments of the same degree as a. It

follows that z =∑i−1

j=1 ajej − aei is a homogeneous cycle of degree > 1. By our assumptionz is a boundary of K(x1, . . . , xi;M), which proves that a ∈ (x1, . . . , xi−1)M.

Lemma 12.8. Let S and M be as in 12.7. Assume that S0 is local with infinite residuefield and assume that Hj(S+;M)k = 0 for j > 0 and k 6= j. Then S+ can be generated by ad-sequence with respect to M.

Proof. By 12.7 it suffices to construct a sequence x1, . . . , xn of 1-fomrs generating S+ suchthat

Hj(x1, . . . , xi;M)k = 0 for j > 0, k 6= j and all i = 1, . . . , n. (3)

Since the residue field of S0 is infinite, we can choose a generating sequence x1, . . . , xnsuch that Hj(x1, . . . , xi;M)k = 0 for j > 0, k >> 0 and all i = 1, . . . , n.

We naow show by induction on ` = n− i that this sequence even satisfies (3). If ` = 0,then this is the assumption of the lemma. Suppose now that ` > 0 and consider the longexact sequence

. . .→ Hj+1(x1, . . . , xi+1;M)k+1 → Hj(x1, . . . , xi;M)kxi+1→ Hj(x1, . . . , xi;M)k+1 →

Hj(x1, . . . , xi+1;M)k+1 → . . .

(here i = n− `).By the induction hypothesis we have Hj(x1, . . . , xi+1;M)k = 0 for k 6= j. It fol-

lows that Hj(x1, . . . , xi;M)k ' Hj(x1, . . . , xi;M)k+1 for k > j. However, by the choiceof the sequence, we have Hj(x1, . . . , xi;M)k = 0 for k >> 0. Thus we conclude thatHj(x1, . . . , xi;M)k = 0 for k > j. Since, trivially, Hj(x1, . . . , xi;M)k = 0 for k < j, theassertion follows.

Theorem 12.9. Let (R,m, k) be a local ring with infinite residue field. Let I be an ideal ofR and M a finitely generated R-module.

(1) The following conditions are equivalent:

(a) I is generated by a d-sequence with respect to M.

46

(b) I is generated by a relative regular sequence with respect to M and is of lineartype with respect to M.

(c) I is generated by a proper sequence with respect to M and is of linear type withrespect to M.

(d) M(I;M) is acyclic.

(2) The following conditions are equivalent:

(a) I is generated by a proper sequence with respect to M.

(b) Z(I;M) is acyclic.

Proof. The implications (1) (a)⇒(b)⇒(c)⇒(d) and (2) (a)⇒(b) follow from our previousconsiderations. It remains to show (1) (d)⇒(a) and (2) (b)⇒(a).

The acyclicity ofM(I;M), resp Z(I;M), may be used to compute H(S+; grI(M)) andH(S+; Sym(I;M)). (By Sym(I;M) we mean H0(Z(I;M)).) From 11.8 we know that

Hi(S+; grI(M))j =

{0 for j 6= i

Hi(I;M) for j = i

Similarly we have, for the Z-complex,

Hi(S+; Sym(I;M))j =

{0 for j 6= i

Zi(I;M) for j = i

Using 12.8 we see that the irrelevant ideal of grI(R) can be generated by a d-sequencewith respect to grI(M). The same holds true for the irrelevant ideal of Sym(I) with respectto Sym(I;M). The proof of the theorem is completed with the next result.

Theorem 12.10. Let x1, . . . , xn be a sequence generating the ideal I and let x1, . . . , xn,resp. x∗1, . . . , x

∗n, be the corresponding sequence generating the irrelevant ideal of grI(R),

resp. Sym(I).

(1) The following conditions are equivalent:

(a) {x1, . . . , xn} is a d-sequence with respect to M.

(b) {x1, . . . , xn} is a d-sequence with respect to grI(M).

(2) ([27, 2.15])The following conditions are equivalent:

(a) {x1, . . . , xn} is a proper sequence with respect to M.

(b) {x∗1, . . . , x∗n} is a d-sequence with respect to Sym(I;M).

Proof. (1) (a)⇒(b) (cf. [22]): By 12.7 it suffices to show that Hi(x1, . . . , xn; grI(M))j = 0for i = 1 . . . , n and j 6= i. Let x =

∑ik=1 akek be a homogeneous cycle of degree j > 1,

i.e. ak ∈ Ij−1M/IjM. Let ak ∈ Ij−1M be representatives of the ak. We then have ai ∈(x1, . . . , xi−1)I

j−1M : xi ∩ Ij−1M + IjM. Since x is a d-sequence we conclude from 12.2that ai ∈ (x1, . . . , xi−1)I

j−2M + IjM. In other words, ai ∈ (x1, . . . , xi−1)grI(M)j−2. Thusthe cycle z is homologous to a cycle

∑i−1k=1 ak

′ek. By induction on i the assertion follows.

47

(b)⇒(a): We show that {x1, . . . , xn} satisfies

((x1, . . . , xi)IM : xi+1) ∩ IkM = (x1, . . . , xi)Ik−1M

for i = 1, . . . , n− 1 and all k ≥ 1. By 12.1(c), it will follow that x is a d-sequence.Since H1(x1, . . . , xi; grI(M))k = 0 for k > 1, the natural map

H0(Fk+1K(x1, . . . , xi;M))→ H0(FkK(x1, . . . , xi;M))

is injective. This says that

(x1, . . . , xi)Ik−1M ∩ Ik+1M = (x1, . . . , xi)I

kM

for i = 1 . . . , n− 1 and k > 1. By induction on k one derives from this equality

(x1, . . . , xi)IM ∩ Ik+1 = (x1, . . . , xi)IkM for all k ≥ 0 (4)

Now let a ∈ (x1, . . . , xi)IM : xi+1∩IkM ; then by (4) axi+1 ∈ (x1, . . . , xi)IM∩Ik+1M ⊂(x1, . . . , xi)I

kM, and thus axi+1 =∑i

j=1 bjxj , bj ∈ IkM. Let a = a + Ik+1M, bj = bj +

Ik+1M ; then axi+1 =∑i

j=1 bjxj and∑i

j=1 bjej − aei+1 is a homogeneous cycle of degree> 0. By 12.7 it is therefore a boundary in K(x1, . . . , xi; grI(M)), which implies that a ∈(x1, . . . , xi)I

k−1M + Ik+1M. Write a = a′ + a′′, a′ ∈ (x1, . . . , xi)Ik−1M, a′′ ∈ Ik+1M. We

have a′′xi+1 ∈ (x1, . . . , xi)Ik−1M ∩ Ik+2M, hence as before a′′ ∈ (x1, . . . , xi)I

kM + Ik+2M.It follows that a ∈ (x1, . . . , xi)I

k−1M + Ik+2M.Repeating this argument we see that

a ∈⋂r≥1

((x1, . . . , xi)Ik−1M + Ik+r) = (x1, . . . , xi)I

k−1M

as required.(2) (a)⇒(b): From the presentation

Z1(I;M)→ Z0(I;M)→ Sym(I;M)→ 0

it follows immediately that H1(x∗; Sym(I;M))j = 0 for j > 1. Suppose we knew that the

natural mappings

γi : H1(x∗1, . . . , x

∗i ; Sym(I;M))→ H1(x

∗; Sym(I;M))

are all injective for i = 1, . . . , n − 1. We would then have H1(x∗1, . . . , x

∗i ; Sym(I;M))j = 0

for i = 1, . . . , n− 1 and j > 1, which by 12.7 implies that x∗ = {x∗1, . . . , x∗n} is a d-sequencewith respect to Sym(I;M).

The injectivity of the maps γi follows once we have shown that ei+1H1(e1, . . . , ei; Sym(I;M)) =0 for i = 1, . . . , n− 1, or, equivalently, that

ei+1(U ∩ (e1, . . . , ei)Z0(I;M))j ⊂ ((e1, . . . , ei)U)j+1

for i = 1, . . . , n− 1 and all j ≥ 1. Here U denotes the image of ∂′ : Z1(I;M)→ Z0(I;M).We prove the above inclusion by induction on j. For j = 1 it amounts to showng that

ei+1Z1(x1, . . . , xi;M) ⊂ (e1, . . . , ei)U,

48

since (U ∩ (e1, . . . , e− i)Z0(I;M))1 = Z1(x1, . . . , xi;M). (Here Z1(x1, . . . , xi;M) is under-stood to be a submodule of Z0(I;M).)

To prove this inclusion, let z ∈ Z1(x1, . . . , xi;M); then xi+1z = ∂u, u ∈ K2(x1, . . . , xi;M),since we assume that x is a proper sequence with respect to M. In the Z-complex we have∂(ei+1 ⊗ z − ∂′(u⊗ 1)) = 0. This shows that

ei+1 ⊗ z − ∂′(u⊗ 1) ∈ Z1(I;M)⊗R[e1, . . . , ei]

and it follows that

ei+1z = ∂′(ei+1 ⊗ z − ∂′(u⊗ 1)) ∈ (e1, . . . , ei)U,

as required.Now let j > 1 and z ∈ (U ∩ (e1, . . . , ei)Z0(I;M))j . We can write z =

∑nk=1 ekz

′k,

with z′k ∈ Uj−1, since U is generated by elements of degree 1. It follows that enz′n ∈

(e1, . . . , en−1)Z0(I;M). Since e1, . . . , en is a regular sequence on Z0(I;M) = M⊗R[e1, . . . , en],we then see that z′n ∈ (e1, . . . , en−1)Z0(I;M).

We thus have z′n ∈ (U ∩ (e1, . . . , en−1)Z0(I;M))j−1 and by the induction hypothesis weknow that enz

′n ∈ ((e1, . . . , en−1)U)j . Therefore we can write z =

∑n−1k=1 e−kz′′k , z′′k ∈ Uj−1.

By induction we obtain a presentation z =∑i

k=1 ekzk with zk ∈ Uj−1, so that ei+1z ∈((e− 1, . . . , ei)U)j+1.

(b)⇒(a): We have the following identifications:

Kj(x1, . . . , xi;M) = Kj(x∗1, . . . , x

∗i ; Sym(I;M))j

andZj(x1, . . . , xi;M) = Hj(x

∗1, . . . , x

∗i ; Sym(I;M))j .

Using them we have that for z ∈ Zj(x1, . . . , xi;M), xi+1z = x∗i+1z ∈ Bj(x∗1, . . . , x∗i ; Sym(I;M))j+1,since x∗ is assumed to be a d-sequence. Hence there exists u ∈ Kj+1(x

∗1, . . . , x

∗i ; Sym(I;M))j+1 =

Kj+1(x1, . . . , xi;M) with xi+1z = x∗i+1z∂u. This proves that x is a proper sequence.In the proof of 12.10, (2) (a)⇒(b) we used only that portion of the definition of a proper

sequence which refer to j = 1 (cf. §6). Thus we obtain

Corollary 12.11. ([27, 2.16]) The following conditions are equivalent:

(a) {x1, . . . , xn} is a proper sequence with respect to M.

(b) xi+1H1(x1, . . . , xi;M) = 0 for i = 1, . . . , n− 1.

13 Linear resolutions

In this section we are mainly concerned with the study of the M-complex associated withthe maximal ideal of a local ring. Assume for new that (R,m, k) is a regular local ring andthat M is a finitely generated R-module. We are going to study the complex

M(m,M) : 0→ Hn(m;M)⊗ S[−n]→ . . .→ H1(m;M)⊗ S[−1]→ H0(m,M)⊗ S → 0

constructed with respect to a minimal system of generators {x1, . . . , xn} of m.

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Recall that S = R[e1, . . . , en and that M(m;M)i = Hi(m;M) ⊗R R[e1, . . . , en] 'k[e1, . . . , en]bi , where bi = dimkHi(m;M).

Thus M(m;M) may be considered to be a complex of free grm(R)-modules, sincegrm(R) ' k[e1, . . . , en], where the isomorphism is given by xi + m2 → ei, for i = 1, . . . , n.

To M we may assign a different complex of free grm(R)-modules: Let

(G, d) : . . .→ G2d2→ G1

d1→ G0 → 0

be a minimal free R-resolution of M. We define a natural filtration on (G, d) by setting

FjGi =

{Gi for j < i

mj−iGi for j ≥ i

The associated graded complex

gr(G) : . . .→ gr(G2)gr(d2)→ gr(G1)

gr(d1)→ gr(G0)→ 0

is then a complex of free grm(R)-modules.

Theorem 13.1. M(m;M) ' gr(G).

Proof. We proceed by induction on pd(M), the projective dimension of M. If M is free,the asset ion is clear. Suppose now pd(M) > 0, and consider the exact sequence

0→ N → G0 →M → 0, N = d1(G1).

Since G0 is free and N ⊂ mG0, we get isomorphisms

Hi+1(m;M) 'σ Hi(m;N), for all i ≥ 0.

Here σ is the connecting homomorphism in the long exact homology sequence.

One checked readily that for all i ≥ 0 the following diagram commutes:

Hi+2(m;M)⊗ S ∂′ //

σ⊗id��

Hi+1(m;M)⊗ S

σ⊗id��

Hi+1(m;M)⊗ S ∂′ // Hi(m;M)⊗ S

Using this fact and the induction hypothesis, all that remains is to prove that

gr(G1)gr(d1)→ gr(G0)

is isomorphic to

H1(m;M)⊗ S ∂′→ H0(m;M)⊗ S.

To see this we chose bases (fk), k = 1, . . . , b1 of G1 and (g`), ` = 1, . . . , b0 of G0. Wethen have d1(fk) =

∑αk`g`, where each αk` can be written as αk` =

∑αik`xi. If we now

put aki =∑αik`g`, then d1(fk) =

∑akixi.

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Let f∗k , resp. g∗, denote the initial forms of fk, resp. g`, in gr(G) and denote by r theresidue class modulo m of an element r ∈ R.

We then obtain

gr(d1)(f∗k ) =

∑`

(∑i

αik`)g∗ (recall that ei = xi + m2).

In order to describe ∂′ we observe that the elements [∑

i ρ(aki)ei], k = 1, . . . , b1 form abasis of H1(m;M) and the elements [ρ(g`)], ` = 1, . . . , b0 form a basis of H0(m;M). (Herethe brackets [ ] denote the homology class of a cycle and ρ : G0 →M is the augmentationmap.)

We now compute ∂′ explicitly:

∂′[∑i

ρ(aki)ei] =∑i

[ρ(aki)]⊗ ei =∑i

∑`

αik`[ρ(g`)]⊗ ei =∑`

[ρ(g`)]⊗∑i

αik`ei.

The assertion follows.

Definition 13.2. The R-module M has a linear resolution if the associated complexgr(G) is acyclic.

Corollary 13.3. (a) If the R-module M has a linear resolution, so does mM.

(b) For an R-module, there exists an integer n0 so that mnM has a linear resolution forn ≥ n0.

Proof. Both assertions follow from the remark following 11.4, 12.8 and the theorem above.

Combining 13.1 and 12.9 we get

Corollary 13.4. Let (R,m) be a regular local ring with infinite residue field, and let M bea finitely generated R-module. The following conditions are equivalent:

(a) The maximal ideal m is generated by a d-sequence with respect to M.

(b) M has a linear resolution.

If the equivalent conditions hold, then

M(m;M) : 0→ Hn(m;M)⊗ S[−n]→ . . .→ H0(m;M)⊗ S → 0

is a minimal free homogeneous grm(R)-resolution of grm(M), and the Hilbert function HM (z)of M is given by the formula

HM (z) = P (z)/(1− z)n, P (z) =∑

(−1)i+1bizi, bi = dimkHi(m;M).

Moreover, depth M = depth grm(M).

Now let (R,m, k) be any local ring (not necessarily regular). We are going to give adescription ofM(m;R). The homology ofM(m;R) does not change if we pass to the m-adiccompletion of R. hence we way assume that R is complete and thus choose a presentationR = A/I, where (A, n, k) is a regular local ring and I ⊂ n2.

let G be a minimal free resolution of R as an A-module.

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Corollary 13.5. Hi(M(m;R)) ' Hi(gr(G)).

Example 13.6. (a) Let I ⊂ k[xij ] be the ideal generated by the maximal minors of an m×matrix of indeterminates. Let A = k[[xij ]], R = A/IA. since the resolution of this ideal isknown to be linear ([8]), we see that

Hi(M(m;R)) = 0 for i > 1, and

H1(M(m;R)) = I.

(b) Let I ⊂ k[xij ] be the ideal generated by 2n × 2n Pfaffians of a skew-symmetric(2n + 1) × (2n + 1) matrix in the indeterminates xij . Again let A = k[[xij ]], R = A/IA.Using the structure theorem of [?], we see that

H1(M(m;R)) = I,

H2(M(m;R)) = 0, and

H3(M(m;R)) = grm(R)

This example shows that, theM-complex is not rigid, in contrast to the ordinary Koszulcomplex.

Quite generally we have that M1(m;R) → M0(m;R) is the zero mapping. Hence thefollowing conditions are equivalent:

(a) H1(M(m;R)) = 0.

(b) M(m;R) is acyclic.

(c) R is a regular local ring.

Thus for a non-regular local ring (R,m), we cannot have much better exactness ofM(m;R)than Hi(M(m;R)) = 0 for i > 1.

Bringing in 12.7, we have

Theorem 13.7. Let (R,m) be a local ring with infinite residue field. The following condi-tions are equivalent:

(a) m is generated by a proper sequence.

(b) Hi(M(m;R)) = 0 for i > 1, and the canonical homomorphism H1(M(m;R)) →Sym(m) is injective.

Suppose again that R has a presentation R = A/I, with (A, n, k) regular, I ⊂ n2, and kinfinite. In general grn(I) 6= ker (grn(A)→ grm(R)). Making an extra assumption on grn(I)we get

Theorem 13.8. Suppose that there exists an integer d ≥ 1 such that grn(I)[−d] = ker (grn(A)→grm(R)). Then the following conditions are equivalent:

(a) The maximal ideal m of R is generated by a proper sequence.

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(b) I has a linear resolution.

If the equivalent conditions hold, then

0→ Hn(m;R)⊗ S[−n− d+ 1]∂′→ . . .

∂′→ H1(m;R)⊗ S[−d]→ H0(m;R)⊗ S → 0 (5)

is a minimal free homogeneous grn(A)-resolution of grm(R), and depth R = depth grm(R).

Remark 13.9. What the hypothesis on grn(I) means is that

I ∩ nν =

{I for ν < d

nν−dI for ν ≥ d.

In other words, I is generated minimally by a standard basis f1, . . . , fm of degree ν(fi) = dfor all i.

Proof. (of 13.8) (a)⇒(b): Let G be a minimal free A-resolution of R. By 13.5 we haveHi(M(m;R)) ' Hi(gr(G)), and by 13.7 we know that Hi(M(m;R)) = 0 for i > 1. Thisshows that

. . .→ gr(G2)→ gr(G1)→ 0

is acyclic, that is, I has a linear resolution.

(b)⇒(a): If I has a linear resolution then arguing as above we get that Hi(M(m;R)) = 0for i > 1 and Hi(M(m;R)) = grn(I). Using the isomorphism

grn(I)[−d] = ker (grn(A)→ grm(R))

we get that the complex (5) is a homogeneous grn(A)-resolution of grm(R).

As an application of these methods we give a generalization of a result of Steurich ([45]),which itself represented an extension of results of [38] and [?].

Theorem 13.10. Let (A, n) be a regular local ring with infinite residue field, and let R =A/I and denote by m its maximal ideal. Assume that for some d ≥ 2 we have:

(i) I ⊂ nd.

(ii) There exists a system of parameters x of R which forms a relative regular sequence,such that md ⊂ (x)md−1.

Then I has a linear resolution.

Proof. x forms part of a minimal system of generators of m and can be completed to onesuch: {x1, . . . , xn, y1, . . . , ym}.

We claim that this sequence is proper. According to the result of Kuhl ??, it suffices toshow that

(a) mH1(x1, . . . , xi;R) = 0 for i = 1, . . . , n;

(b) mH1(x1, . . . , xn, y1, . . . , yj ;R) = 0 for j = 1, . . . , n.

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(a) follows from (b), since we assume that x is a relative regular sequence and , by (b),mH1(x1, . . . , xi;R) = 0.

To prove (b), choose z ∈ Z1(x,y;R), z =∑

i aiei +∑

j bjfj , ∂ej = xi, ∂fj = yj . Since

I ⊂ nd we conclude that all ai, bj ∈ md−1. For x ∈ m, xbj ∈ md = (x)md−1 and thusxbj =

∑bjixi with bji ∈ md−1. It follows that z − ∂(

∑i,j bijei ∧ fj) =

∑a′iei where all

a′i ∈ md = (x)md−1. Since x is a relative regular sequence it follows that xz is a boundary.Next we show that I is generated minimally by a stander basis g1, . . . , gr, with ν(gi) =

degree(gi) = d.Let K = K(m;R), F the m-adic filtration on K. Our proof thus far shows that H1(FiK) =

H1(K) for i ≤ d and H1(Fd+1K) = 0. This shows that H1(K(m∗; grm(R))i = 0 for i 6= d,and H1(m;R) ' H1(FdK) ' H1(K(m∗; grm(R))i, where m∗ is the irrelevant maximal idealof grm(R).

This shows that I∗ = ker (grn(A)→ grm(R)) is generated by forms of degree d. In otherwords, I has a standard basis g1, . . . , gr, with degree(gi) = d, generating I minimally sinceν(I) = dimH1(m;R) = dimH1(m

∗; grm(R)) = ν(I∗).

References

[1] M. Auslander and D. Buchsbaum, Codimension and Multiplicity, Annals of Math. bf68 (1958), 625–657; Corrections, bf 70 (1959), 395–397.

[2] L. Avramov and J. Herzog, The Koszul algebra of a codimension 2 embedding, Math.Zeit. 175 (1980), 249–260.

[3] Y. Aoyama, A remark on almost complete intersections, Manuscripta Math. Zeit. 22(1977), 225–228.

[4] H. Bass, Injective dimension in Noetherian rings, Trans. Amer. Math. Soc. 102 (1961),18–29.

[5] D, Buchsbaum and D. Eisenbud, What makes a complex exact?, J. Algebra 25 (1973),259-268.

[6] D, Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions, and somestructure theorems for ideals of codimension 3, Amer. J. Math. 99 (1997), 447–485.

[7] M. Brodmann, A Macaulayfication of unmixed domains, J. Algebra 44 (1977), 221-234.

[8] J. Eagon and D. G. Northcott, Ideals defined by matrices and a certain complex asso-ciated with them, Proc. Royal Soc. 269 (1962), 188–204.

[9] D. Eisenbud and C. Huneke, Cohen–Macaulay Rees algebras and their specializations,preprint, 1980.

[10] G. Faltings, Uber Macaulayfizierung, Math. Ann. 238 (1978),175–192.

[11] M. Fiorentini, On relative regular sequences, J. Algebra 18 (1971), 384–389.

[12] T. Gulliksen and G. Levin, Homology of Local Rings, Queen’s Paper in Pure andApplied Math. 20, Queen’s University, Kingston, 1969.

54

[13] J. Herzog, Ein Cohen–Macaulay Kriterium mit Anwendungen auf den Konormalen-modul und den Differentialmodul, Math. Zeit. 163 (1978), 149–162.

[14] J. Herzog, Komplexe, Auflsungen und Dualitat in der lokalen Algebra, Habilitationss-chrift, Universitat Regensburg, 1974.

[15] J. Herzog and E. Kunz, Der kanonische Modul eines Cohen–Macaulay Rings, LectureNotes in math. 238, Springer, Heidelberg, 1971.

[16] M. Hochster and L. J. Ratliff, Jr., Five theorems on Macaulay rings, Pacific J. Math.44 (1973), 147–172.

[17] M. Hochster, Criteria for equality of ordinary and symbolic powers of an ideal, Math.Zeit. 133 (1973), 53–65.

[18] J. Herzog, A. Simis and W. V. Vasconcelos, Approximation complexes and blowing–uprings I, J. Algebra, to appear.

[19] J. Herzog, A. Simis and W. V. Vasconcelos, Approximation complexes and blowing–uprings II, preprint, 1981.

[20] C. Huneke, The theory of d-sequences and power of ideals, Adv. in Math. to appear.

[21] C. Huneke, On the symmetric and Rees algebras of an ideal generated by a d-sequence,J. Algebra 62 (1980), 268–275.

[22] C. Huneke, Symbolic powers of prime ideals and special graded algebras, Comm. inAlgebra 9 (1981), 339–366

[23] C. Huneke, The Koszul homology of an ideal, preprint, 1981.

[24] C. Huneke. Linkage and the Koszul homology of ideals, preprint, 1981.

[25] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.

[26] E. Kunz, The conormal module of an almost complete intersection, Proc. Amer. Math.Soc. 73 (1979),15–21.

[27] M. Kuhl, Uber die Symmetrische Algebra eines Ideals, Dissertation, Universitat Essen,1981.

[28] K. Lebelt, Freie Auflosungen auβerer Potenzen, Manuscripta Math. 21 (1977), 341–355.

[29] H. Matsumura, Commutative Algebra, Benjamim-Cummings, New York, 1980.

[30] T. Matsuoka, On almost complete intersections, Manuscripta Math. 21 (1977), 329–340.

[31] A. Micali and N. Roby, Algebres symetriques et syzygies, J. Algebra 17 (1971), 460–469.

[32] A. Micali, Sur les algebres universelles, Ann. Inst. Fourier, 14 (1964), 33–88.

55

[33] A. Micali, P. Salmon and P. Samuel, Integrite et factorialite des algebres symetriques,Atas IV Coloquio Brasileiro de Matematica, 61–67, Sao Paulo, 1965.

[34] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, IHES Publ.Math. 42 (1973), 323–395

[35] C. Peskine and L. Szpiro, Liaison des varietes algebriques. Inventiones Math. 26 (1973),271–302.

[36] L. J. Ratliff, Jr., On the prime divisors of zero in form rings, Pacific J. Math. 70 (1977),488–517.

[37] M. E. Rossi, On symmetric algebras which are Cohen–Macaulay, Manuscripta Math.34 (1981), 199–210.

[38] J. Sally, Cohen–Macaulay rings of maximal embedding dimensions, J. Algebra 56(1979), 168–183.

[39] P. Salmon, Sulle algebre graduate relative ad un ideale, Symposia Mathematica 8(1972), 269–293.

[40] P. Schenzel, Uber die freien Auflosungen extremaler Cohen–Macaulay-Ringe, J. Alge-bra 64 (1980), 93–101.

[41] J. P. Serre, Algebre Locale-Multiplicites, Lectures Notes in Math., 11, Springer, Hei-delberg, 1965.

[42] A. Simis, Koszul homology and its syzygy-theoretic part, J. Algebra 55 (1978), 28–42.

[43] A. Simis and W. V. Vasconcelos, The syzygies of the conormal module, Amer. J. Math.103 (1981), 203–224.

[44] A. Simis and W. V. Vasconcelos, On the dimension and integrality of symmetric alge-bras, Math. Zeit. 177 (1981), 341–358.

[45] M. Steurich, On rings with linear resolutions, J. Algebra, to appear.

[46] G. Valla, On the symmetric and Rees algebras of an ideal, Manuscripta Math. 30(1980), 239–255.

[47] W. V. Vasconcelos, On the homology of I/I2, Comm. in Algebra 6 (1978), 1801–1809.

[48] W. V. Vasconcelos, The conormal bundle of an ideal, Atas V Escola de Algebra, 111–166, IMPA, Rio de Janeiro, 1979.

[49] J. Watanabe, A note on Gorenstein rings of embedding codimension 3, Nagoya Math.J. 50 (1973), 227–232.

[50] J. Weyman, Resolutions of the exterior and symmetric powers of a module, J. Algebra58 (1979), 333–341.

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