QUASI-COMPLETE INTERSECTIONS WITH

QUASI-COMPLETE INTERSECTIONS WITH APPLICATIONS TO FREE RESOLUTIONS OVER ARTINIAN RINGS

by

Inês B. Henriques

A THESIS

Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Master of Science

Major: Mathematics

Under the Supervision of Professor Luchezar L. Avramov

Lincoln, Nebraska May, 2010

ii

Contents

Contents

ii

1 Introduction

1

2 Exact zero divisors

9

2.1

Exact zero divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2

Deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.3

Betti numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.4

Poincaré series of graded modules . . . . . . . . . . . . . . . . . . . .

30

3 Exact zero divisors over Artinian Gorenstein rings

34

3.1

Short local rings with balanced Hilbert series . . . . . . . . . . . . . .

35

3.2

Modules over short local rings . . . . . . . . . . . . . . . . . . . . . .

41

3.3

Short local Gorenstein rings . . . . . . . . . . . . . . . . . . . . . . .

52

Bibliography

59

1

Chapter 1 Introduction This dissertation contains this introduction and two additional chapters devoted to the proofs of our main results, it lays within the subject area of commutative homological algebra. Commutative algebra had its origins in the study of systems of polynomial equations in several variables and has developed greatly, especially since the introduction of homological methods. We devote our attention to the study of free resolutions of modules over specific classes of rings. Free resolutions are objects that arise from systems of linear equations over rings. It is well known from linear algebra, that systems of linear equations over fields, such as the real numbers or the complex numbers, have two basic properties. Every solution of such a system is a linear combination of a finite set of solutions, which can be chosen to be linearly independent. Auslander, Buchsbaum and Serre established that independence of the spanning set of solutions is obtained in a finite number of steps for the class of regular rings. Zariski established a tight connection between regular rings and singularities; namely, that regular local rings correspond to smooth points on an algebraic variety. Important information on the structure of free resolutions and the module can

2 be obtained from the generating function on the ranks of the maps in the resolution called the Betti numbers. This formal power series is called the Poincaré series of the module M and defined by

PR M (t)

=

∞ X

n rankk TorR n (M, k)t .

n=0

Generating functions are classically used in commutative algebra, another important example is the Hilbert series of a module. The Hilbert series is the formal power series ∞ X

mn M HM (t) = rankk tn . n+1 m M n=0 Although, the Hilbert series and the Poincaré series of a finitely generated Rmodule behave very differently and encode different invariants of the module, there is a powerful interplay between the two generating series. The asymptotic behavior of their coefficients is quite different. In general, the Betti numbers of a module tend to have exponential growth whereas the coefficients of the Hilbert series are asymptotically given by a polynomial. All Hilbert series are known to be rational functions with denominator (1 + t)dim R whereas Poincaré series may be transcendental. Rationality of a generating function is equivalent to the existence of a linear recursive relation with constant coefficients for its originating sequence. In the case of Poincaré series, one obtains a linear recursive relation among the Betti numbers. Hence providing precise and explicit information on the resolution. The tight connection between the two generating functions is characteristic of Koszul modules. A module is said to be Koszul if the maps in its minimal resolutions are given by matrices of linear forms. The Poincaré series of a Koszul module is a

3 rational function with denominator given by the Hilbert Series. Another common technique in the study of commutative Noetherian rings is the reduction to the residue ring modulo the ideal generated by a nonzero element a. Let R denote the residue ring R/aR. Taking the residue ring modulo a nonzero element a corresponds to adding a relation among the generators of R. In this way, the ring R, although smaller than R, may be much more complex. In general, it is not possible to control the behavior of structural or homological properties of modules under the addition of a new relation. The change of rings R → R/aR as been carefully traced in the case where a is a nonzero divisor. Important change of rings results include the comparative study of their Krull dimension, depth, homological dimension, as well as their Poincaré, Bass and Hilbert series. By considering successive quotients by nonzero divisors one can descend to a ring of zero depth. An important case is the reduction to an Artinian ring, for Cohen-Macaulay rings. Note that nonzero divisors are characterized by a trivial annihilator; a is a zero divisor whenever the R/aR-module (0 :R a) is nonzero. Homologicaly, the simplest case to consider is when the annihilator is a free module of positive rank; Blanco (cf [44]) showed that this condition is equivalent to (0 :R a) ∼ = R/aR. In Chapter 2 we study this special class of zero divisors that we henceforth refer to as exact zero divisors; we say that a is an exact zero divisor if 0 6= (0 :R a) ∼ = R/aR 6= 0. While the terminology has been introduced in the recent paper [33], the concept has surfaced the literature in work of André [2] and Soto [27], [44]. The defining property has also appeared in the context of non-commutative rings, cf. Nicholson [39]. We are mainly interested on the descent and ascent of homological and structural properties of the change of rings given by factoring out an exact zero divisor. Important results on change of rings modulo an exact zero divisor include the comparative study of

4 Krull dimension, depth, type, homological dimensions and free resolutions of modules over these rings. Somewhat surprisingly, exact zero divisors share many of the properties of nonzero divisors. In fact, both are instances of the more general concept of quasi-complete intersection homomorphisms which are defined in terms of Koszul homology. A particularly interesting property of this type of zero divisors is that they preserve invariants whose extreme cases characterize the regularity properties in the following chain of implications:

Complete Intersection ⇒ Gorenstein ⇒ Cohen-Macaulay.

We now give precise statements of the main research results contained in this thesis. The descent of the Gorenstein property modulo an exact zero divisor follows from a collapsing spectral sequence yielding isomorphisms of the Ext modules over the respective rings. The descent of the Cohen-Macaulay and Gorenstein properties as well as the ascent of the latter had been proved using a different approach by Rodicio and Soto in [27]. We were able to conclude the following result: Theorem A. If R is a local Noetherian ring and a is an exact zero divisor, then 1. R is a complete intersection if and only if R/aR is a complete intersection. 2. R is a Gorenstein if and only if R/aR is a Gorenstein. 3. R is a Cohen-Macaulay if and only if R/aR is a Cohen-Macaulay. The ascent and descent of the Cohen-Macaulay and complete intersection properties modulo an exact zero divisor follow from the preservation of the corresponding defect invariants: ci-defect and CM-defect.

5 We define the Cohen-Macaulay defect, the complete intersection defect and type of a local ring (S, n, k) as follows: cmd S = dim S − depth S, 1 βS (k) + 1 2 cid S = βS (k) − + dim S, 2

(1.0.0.1)

type S = rankk rankk ExtSdepth S (k, S). These are non-negative invariants of the ring. As shown in [10], cid S = 0 if and only if S is a complete intersection ring; that is, the m-adic completion of S is isomorphic to a quotient ring of a regular local ring by an ideal generated by a regular sequence. In a similar way, the vanishing of cmd(S) characterizes the Cohen-Macaulay property, and S is Gorenstein if in addition, type S = 1. The type of a ring is part of a family of invariants called Bass numbers; the nth Bass number of the ring S, denoted µnS or µnS (k), is defined to be µnS (k) = rankk ExtSn (k, S). Observe that depth S = min{n ≥ 0k µnS (k) 6= 0}. The following theorem summarizes the persistence of these invariants modulo an exact zero divisor: Theorem B. Let R be a commutative local ring and a ∈ m an exact zero divisor. Set R = R/aR. Then the following hold: (1) dim R = dim(R) (2) µiR (k) = µiR (k) for all i > 0. In particular, depth R = depth(R) and type R = type(R).

6 (3) If a ∈ m2 , then βiR (k) = βiR (k) for all i > 0. (4) If a ∈ / m2 , then βiR (k) =

Pi

j=1

βjR (k) for all i > 0.

We observe that (2) was already proved in [44, Prop.2] and (4) recovers Theorem 1.7 in [33]. The following corollary follows from (1.0.0.1) and Theorem 1 generalizes parts (3) and (4) of Theorem 1. Corollary C. cmd(R) = cmd(R) and cid(R) = cid(R). Theorem D. Assume a is an exact zero divisor. M (1) If a ∈ m Ann M , then PM R (t) = PR (t). R −1 (2) If a ∈ / m2 , then PR M (t) = PM (t)(1 − t)

In the subsequent chapter, we apply this knowledge to the study of minimal free resolutions of finitely generated modules over artinian local rings, by means of their Poincaré series. In chapter 3 we identify a class of local rings R with m4 = 0 and the property that HR (−t) PR M (t) is a polynomial in Z[t] for every finite R-module M ; in particular, PR M (t) is a rational function in t. Research on the rationality of Poincaré series has uncovered the existence of both bad and good behavior. Anick [3] constructed the first example of a local ring R for which PR k (t) is irrational. Jacobsson [34] produced a local ring R and finite R-module R M such that PR k (t) is rational, but PM (t) is irrational. More recently, Roos [40]

constructed a local ring R and a family of finite R-modules {Mi }i≥0 such that the series PR Mi (t) is rational for all i ≥ 0, but do not admit a common denominator. Remarkably, in all these examples the rings are short, in the sense that they satisfy m3 = 0. On the other hand, there is a growing gallery of classes of rings R which are

7 good, in the sense of [40], that is, there exists dR (t) ∈ Z[t] such that dR (t) PR M (t) ∈ Z[t] for all finite R-modules M . Among such rings are those with µ(m) − depth R ≤ 3 and the Gorenstein rings R with µ(m) − depth R = 4, cf. [7] and [35], where µ(m) denotes the minimal number of generators of m. All Gorenstein rings (R, m, k) with m3 = 0 are known to be good, due to Sjödin [43]; see also Avramov, Iyengar, S¸ega [8]. However, Gorenstein local rings with m4 = 0 can exhibit bad behavior, as there exist examples with µ(m) ≥ 12 for which PR k (t) is irrational (cf. Bøgvad [16]). The main result, in the primary case of interest, is stated below: Theorem E. Let (R, m, k) be Gorenstein with m4 = 0 and µ(m) = e ≥ 3. If there exists a non-zero element a ∈ m such that the ideal (0 :R a) is principal, then the following hold: (1) HR (t) = 1 + et + et2 + t3 ; (2) HR (−t) PR k (t) = 1; (3) HR (−t) PR M (t) is in Z[t], for every finite R-module M . The hypothesis of our theorem is verified by ‘generic’ Gorenstein rings with m4 = 0 and µ(m) ≥ 3. Indeed, the set of Gorenstein standard graded k-algebras R with HR (t) = 1 + et + et2 + t3 can be parametrized by a projective space by means of the inverse systems correspondence of Macaulay, which associates to each such ring a form of degree 3. With this parametrization, Conca, Rossi and Valla [24] show that a generic Gorenstein graded k-algebra R as above contains a non-zero element a ∈ m for which the ideal (0 :R a) is principal. They also establish (2), which is equivalent to saying that R is a Koszul algebra.

8 While the most significant statement of our results is given for Gorenstein rings, we here work with larger classes of rings. We say that an element a ∈ R is an exact zero divisor if R 6= (0 :R a) ∼ = R/aR 6= 0. Note that an element a satisfying the hypothesis of the Theorem is an exact zero divisor (from the double annihilator property of ideals over Gorenstein rings).

9

Chapter 2 Exact zero divisors Introduction In this chapter we explore the homological and structural traits of the change of rings given by factoring out an exact zero divisor. Among the results reported on change of rings modulo an exact zero divisor, we emphasize the comparative study of Krull dimension, depth, type, homological dimensions and free resolutions of modules over these rings. Note that similar results are also known for the change of rings modulo a regular element although their study resources to substantially different mathematical techniques.

2.1

Exact zero divisors

Let R be a commutative ring and M an R-module. As in [33, §1], we say that an element a ∈ R is an exact zero divisor if it satisfies the conditions R 6= (0 : a) ∼ = R/aR 6= 0.

10 Or, equivalently, a = 6 0 and there exists b 6= 0 satisfying (0 : a) = bR and (0 : b) = aR. We begin by establishing the following theorem: Theorem 2.1.1. Let R be a local ring and a an exact zero divisor in R, one has

dim(R/aR) = dim R ,

depth(R/aR) = depth R ,

cmd(R/aR) = cmd R ,

type(R/aR) = type R .

We precede the proof of the theorem with three simple observations. Remark 2.1.2. For elements a and b in a ring R the following are equivalent: (i) (0 : a) = bR and (0 : b) = aR. (ii) For each R-module M there are isomorphisms

∼ TorR i (R/aR, M ) =

   (0 :

a)/bM

for odd i ≥ 1 ,

  (0 :M b)/aM

for even i ≥ 2 .

M

When they hold, for each R-module M there are isomorphisms     (0 : a)M     ExtiR (R/aR, M ) ∼ = (0 :M b)/aM       (0 :M a)/bM

for i = 0 , for odd i ≥ 1 ,

(2.1.2.1)

for even i ≥ 2 .

Indeed, when (i) holds, the R-module R/aR has a periodic free resolution b

a

b

a

··· → R → − R→ − R→ − R→ − R → 0 → ···

i from which the expressions for TorR i (R/aR, M ) and ExtR (R/aR, M ) follow.

(2.1.2.2)

11 Conversely (ii) implies (i) by taking M = R and noting that TorR i (R/aR, R) = 0 for i ≥ 0. When (R, m, k) is a local Noetherian ring, the equivalence of the conditions above for a ∈ m is also proved by Soto. In addition, he proves that they are also equivalent to the freeness of the R/aR-module (0 :R a); see [44, Prop. 1]. We call any such element b a complementary divisor for a, and say that a, b form an exact pair of zero divisors. Note that if R is local then b is unique up to a unit. 2.1.3. Exact zero divisors under flat homomorphisms. Let ϕ : R → S be a flat ring homomorphism. It follows from the discussion above that the images under ϕ of an exact pair of zero divisors in R form an exact pair of zero divisors in S provided they are non-zero. Example 2.1.4. Let (P, p, k) be a local ring and u, v, w be elements in p. If uv, w is a regular sequence, then the images of u, v in Q = P/(uv, w) form an exact pair of zero divisors, as seen by direct computation. Lemma 2.1.5. Let R be a commutative ring and M an R-module of finite length. For a ∈ R, an equality (0 : a)M = bM implies (0 : b)M = aM . Proof. From a(bM ) = a(0 : a)M = 0 one gets aM ⊆ (0 : b)M . A length count, using this inclusion and the composition M/bM = M/(0 : a)M ∼ = aM , gives

`(M ) = `(M/bM ) + `(bM ) ≤ `(aM ) + `((0 : a)M ) = `(M )

These relations imply `((0 : a)M ) = `(bM ), hence (0 : a)M = bM . The Proposition below gives a more general version of Soto’s results in [44].

12 Proposition 2.1.6. Let M, L be R-modules. If a is an exact zero divisor, aL = 0, and TorR i (R/aR, M ) = 0 for all i > 0, then the following R-module isomorphisms hold: ∗ (M , L) ∼ (1) ExtR = Ext∗R (M, L)

(2) Ext∗R (L, M ) ∼ = Ext∗R (L, M ) R ∼ (3) TorR ∗ (M , L) = Tor∗ (M, L)

where R = R/aR and M = M/aM . Note that if a is an exact zero divisor then Remark 2.1.2 yields the equivalence of the following statements: 1. (0 :M a) = bM and (0 :M b) = aM for some complementary divisor b. 2. TorR i (R/aR, M ) = 0 for all i > 0. Proof of Proposition 2.1.6. For the proof, we will use the classical change-of-ring spectral sequences, see [20]. (a) Consider the spectral sequence

p+q ExtpR (TorR q (R, M ), L) ⇒ ExtR (M, L)

Since TorR q (R, M ) = 0 for all q > 0 (cf. Remark 2.1.2), this spectral sequence degenerates, yielding: p p ExtpR (M, L) ∼ = ExtR (R ⊗R M, L) ∼ = ExtR (M , L)

(b) Consider the spectral sequence

ExtpR (L, ExtqR (R, M )) ⇒ Extp+q R (L, M )

13 Since ExtqR (R, M ) = 0 for all q > 0 (cf. Remark 2.1.2), this spectral sequence degenerates, yielding: p p ExtpR (L, M ) ∼ = ExtR (L, HomR (R, M )) ∼ = ExtR (L, M )

(c) Consider the change-of-rings spectral sequence

R R TorR p (L, Torq (R, M )) ⇒ Torp+q (L, M )

Since TorR q (R, M ) = 0 for all q > 0 (cf. Remark 2.1.2), this spectral sequence degenerates, yielding: R R ∼ ∼ TorR p (L, M ) = Torp (L, R ⊗R M ) = Torp (L, M )

2.1.7. Socle ideal under change of rings.It follows from the Lemma 2.1.8 that the socle of the rings is preserved under change of rings modulo an exact zero divisor. We now give a different reasoning for this fact: Lemma 2.1.8. Let R be a commutative ring and M, N be R-modules. If a ∈ R satisfies aM = 0 and N/aN ∼ = (0 :N a), then there exists an isomorphism of R-modules: HomR/aR (M, N/aN ) ∼ = HomR (M, N ).

In particular, if R is local with maximal ideal m and a is an exact zero divisor, then (0 :R m) ∼ = (0 :R m)

14 where R = R/aR and m = m/aR. Proof. The following R-module isomorphisms are R/aR-linear: N/aN ∼ = (0 :N a) ∼ = HomR (R/aR, N ) .

We have thus: HomR/aR (M, N/aN ) ∼ = HomR/aR (M, HomR (R/aR, N )) ∼ = HomR (M ⊗R/aR R/aR, N ) ∼ = HomR (M, N )

where the second row is given by Hom-Tensor adjointness. The last conclusion is obtained by taking M = R/m and N = R. We derive the following Corollary to Proposition 2.1.6: Corollary 2.1.9. If R is a local ring and a an exact zero-divisor, then

µiR/aR = µiR

for all i ≥ 0 .

Proof of Theorem 2.1.1. The equalities for depth and type follow from Corollary 2.1.9, the equality for Cohen-Macaulay defects from those for depth and dimension. Assume that dim(R/aR) 6= d holds, where d = dim R. Choosing b ∈ R with (0 : a) = bR, we then have dim(R/aR) < dim(R/bR) = d, due to the equality

Spec R = SuppR (R/aR) ∪ SuppR (R/bR) .

15 Thus, every q ∈ Spec R with dim(R/q) = d satisfies a 6∈ q and b ∈ q, so we get

dim(R/aR) ≤ d − 1 = dim(R/(q + aR)) ≤ dim(R/aR)

It follows that both R/aR and R/(q + aR) have dimension d − 1. Choose a prime ideal p in SuppR (R/(q + aR)), for which dim(R/p) = d − 1 holds. The equality dim(R/aR) = d − 1 then implies that p is minimal over aR. After localizing at p and changing notation, we may assume that (R, m, k) is a local ring with dim R = 1 and elements a and b in R, satisfying

(0 : a) = bR ,

(0 : b) = aR ,

dim R/aR = 0 ,

and

dim R/bR = 1 .

One has mh ⊆ aR for some integer h ≥ 1. By the Artin-Rees Lemma, there is an integer i ≥ 0, such that mi ∩ bR = mj−i (mi ∩ bR) holds for all j ≥ i, so we get

mj ∩ bR = mj−i (mi ∩ bR) ⊆ mh b ⊆ Rab = 0 for all j ≥ h + i .

Set Rj = R/mj . The condition for an element c + mj of R/mj to be in (0 : b)Rj is cb ∈ mj ∩ (Rb). For j ≥ h + i this implies cb = 0, by the calculation above. As (0 : b) = aR holds by hypothesis, we get c ∈ aR, hence (0 : b)Rj ⊆ aRj , and thus (0 : b)Rj = aRj . As Rj is artinian, so Lemma 2.1.5 gives (0 : a)Rj = bRj . Now fix some j ≥ h + i, and form the exact sequence of R-modules

0 → mj /mj+1 → R/mj+1 → R/mj → 0

16 By Lemma 2.1.2, the first and last terms in the induced exact sequence

R j j j+1 j+1 TorR ) → TorR ) 2 (R/aR, R/m ) → Tor1 (R/aR, k) ⊗k (m /m 1 (R/aR, R/m

are zero. Also, mj /mj+1 = 6 0 as R is not artinian. Thus, we get TorR 1 (R/aR, k) = 0, hence R/aR is free over R, and thus a = 0. This is the desired contradiction. The theorem implies that when a is an exact zero divisor, R/aR is Cohen-Macaulay if and only if R is. In fact, the following more precise assertion holds: Proposition 2.1.10. Let R be a local ring and a an element of R. The following conditions are equivalent: (i) R/aR is Cohen-Macaulay with dim(R/aR) = dim R and (0 : a) is principal. (i0 ) depth(R/aR) ≥ dim R and (0 : a) is principal. (ii) R is Cohen-Macaulay and a is an exact zero-divisor. Proof. (ii) =⇒ (i0 ). This follows from Theorem 2.1.1. (i0 ) =⇒ (i). Use the inequalities depth(R/aR) ≤ dim(R/aR) ≤ dim R. (i) =⇒ (ii). The hypothesis implies (0 : a) = bR 6= R for some b ∈ R. Set K = (0 : b)/aR and pick any p ∈ AssR (K). The inclusion K ⊆ R/aR shows that p is also in AssR (R/aR). The Cohen-Macaulayness of R/aR implies that p is minimal in SuppR (R/aR) and satisfies dim(R/p) = dim(R/aR). We have dim(R/aR) = dim R, so p is minimal in Spec R, hence the ring Rp is artinian. As one has (0 : (a/1))Rp = (b/1)Rp , Lemma 2.1.5 yields (a/1)Rp = (0 : (b/1))Rp ; that is, Kp = 0. Since p was arbitrary in AssR (K), we conclude that K is equal to zero. Thus, a is an exact zero-divisor. By Theorem 2.1.1, R is Cohen-Macaulay. The hypothesis dim(R/aR) = dim R in (i) cannot be deleted :

17 Example 2.1.11. For R = k[x, y]/(xy, y 2 ) one has (0 : x) = yR, and R/xR ∼ = k[y]/(y 2 ) is Cohen-Macaulay, but x is not an exact zero-divisor, as (0 : y) = (x, y)R. Note that one has depth R/xR = dim R/xR = 0 dim R = 1. The following corollary follows, also in the spirit of André’s result [2]: Corollary 2. Assume a is an exact zero divisor. Any two of the following statements implies the third: (1) R is Gorenstein (2) R is Gorenstein (3) a is an exact zero divisor. Proof. Recall that for local rings, the Gorenstein property is the restriction of the Cohen-Macaulay property to rings of type one. Recall that the type of a ring R is R type(R) = µdepth . In view of Corollary 2.1.9 and Proposition 2.1.10, it suffices to R

assume that (1) and (2) hold. In this case, both R and R/aR are Cohen-Macaulay 2.1.10 implies that (3) holds.

2.2

Deviations

Let (R, m, k) be a local ring with maximal ideal m and residue field k. 2.2.1. DG Γ algebras. A system of divided powers on a DG algebra Λ is an operation that associates to every element a ∈ Λ of even positive degree a sequence of elements a(i) ∈ Λ with i = 0, 1, 2 . . . , satisfying certain axioms, cf. [30, 1.7.1]. A DG Γ- algebra is a DG-algebra with divided powers which are compatible with the differential, in the sense that ∂(a(i) ) = ∂(a)a(i−1) .

18 There exists a free resolution of k over R which has a structure of DG Γ-algebra, obtained by Tate’s procedure of adjoining variables to kill cycles. We call such a resolution a Tate resolution. This resolution can be chosen to be minimal, cf. More generally, a DG Γ algebra Ahxi obtained from a given DG algebra A by adjunction of variables x = x1 , x2 , . . . is called a semi-free Γ extension of A. If Ahxi is a semi-free Γ extension of A, then for each q we denote by Ahx6q i the semi-free Γ extension of A obtained by adjoining all variables x of degree at most q. For every set of cycles z ⊆ A, Tate [45, §1] described a procedure for adjoining a set x = {xz }z∈z , consisting of exterior variables of odd degrees and of divided powers variables in even degrees, resulting in a DG R-algebra Ahxi that contains A as a subalgebra, and where 2.2.1.1. Let ϕ : A → B be a surjective quasi-isomorphism of DG Γ-algebras. If Bhyi is a semi-free Γ extension of B, with y = y1 , y2 , . . . , then there exists a semi-free Γ extension Ahxi of A with x = x1 , x2 , . . . such that ϕ extends to a surjective quasiisomorphism ϕ : Ahxi → Bhyi with ϕ(xi ) = yi for each i and such that the induced map ϕ6q : Ahx6q i → Bhy 6q i is a surjective quasi-isomorphism for each q. The proof of this statement is obtained by repeated application of [30, 1.3.5], along the lines of the proof of [30, 1.9.7]. See also the proof of [, ]. 2.2.2. Indecomposables. Let Λ be a DG Γ-algebra with Λ0 = R and Λi = 0 for i < 0. We denote Λ>0 the ideal of elements of positive degree. Let C(Λ) denote the ideal generated by all elements of the form uv with u, v ∈ Λ>0 and w(n) with w ∈ Λ2i , i > 0, n ≥ 2; this is called the module of decomposables of Λ. The module of Γ-indecomposables of Λ is the quotient of Λ>0 by C(Λ). Note that TorR (k, k) inherits a structure of DG Γ algebra from the minimal Tate resolution of k. We denote π∗ (R) the module of indecomposables of TorR (k, k).

19 If ν : R → S is a surjective homomorphism of local rings, note that ν induces a homomorphism π∗ (ν) : π∗ (R) → π∗ (S) of graded k-vector spaces. Theorem 2.2.3. Let (R, m, k) be a local ring and a ∈ R an exact zero divisor. Set R = R/aR and let ν : R → R denote the canonical projection. Then πi (ν) is an isomorphism for all i ≥ 3. Further, (1) If a ∈ m2 , then π1 (ν) is an isomorphism and one has an exact sequence of k-vector spaces: π2 (ν)

0 → k → π2 (R) −−−→ π2 (R) → k → 0

(2) If a ∈ / m2 , then for i = 1, 2 one has an exact sequences of k-vector spaces: πi (ν)

0 → k → πi (R) −−−→ πi (R) → 0

The proof of the Theorem will be given at the end of the section. We now provide several corollaries, with the necessary preparation. 2.2.4. Deviations. We recall that the deviations εi (R) of the local ring R are defined from the equation of formal power series:

PR k (t)

=

∞ Y (1 + t2i−1 )ε2i−1 (R) i=1

(1 − t2i )ε2i (R)

,

(2.2.4.1)

cf. [10, Remark 7.1.1]); where PR e series of k over R. k (t) denotes, as usual, the Poincar´ The following equalities are known to hold (see [30, Chap.3,§1] ):

dimk πi (R) = εi (R) for all i ≥ 0

(2.2.4.2)

20 Corollary 2.2.5. The following formulas hold:

R PR k (t) = Pk (t) R −1 PR k (t) = Pk (t)(1 − t)

if a ∈ m2 if a ∈ / m2

Proof. It follows from Theorem 2.2.3, that εi (R) = εi (R), for all i ≥ 3. Further, (1) If a ∈ m2 , then εi (R) = εi (R) for i = 1, 2. (2) If a ∈ / m2 , then εi (R) = εi (R) + 1 for i = 1, 2. From this and (2.2.4.1) one can immediately deduce the formulas for the Poincaré series of the residue field. 2.2.6. Complete intersection defect. The complete intersection defect of a ring S, defined in (1.0.0.1), can also be obtained from the following equality:

cid(S) = ε2 (S) − ε1 (S) + dim S

(cf. [19])

Recall that S is a complete intersection if and only if cid(S) = 0. Corollary 2.2.7. If a is an exact zero divisor, then cid(R) = cid(R). In particular, R is a complete intersection if and only if R is a complete intersection. The result of the following Corollary has already been proved, more generally, by André. Corollary 2.2.8. Then any two of the following conditions implies the third: (1) R is a complete intersection. (2) R/aR is a complete intersection.

21 (3) a is an exact zero divisor. Proof. In view of Corollary 2.2.7, it suffices to assume that (1) and (2) hold. In particular, both R and R/aR are Gorenstein. Then 2.1 implies that (3) holds. Definition. A homomorphism of local rings ν : (R, m, k) → (S, m, k) is said to be large if it induces a surjective homomorphism ν : TorR (k, k) → TorS (k, k). The reader may consult [38, 1.1] for additional details on large homomorphisms. Corollary 2.2.9. If a ∈ / m2 then ν : R → R is a large homomorphism. Proof. Theorem 2.2.3(2) gives that πi (ν) is surjective for all i. This implies that ν is large as follows from [11, Corollary 1.3]. Corollary 2.2.10. If a is an exact zero-divisor, then

R −1 PR . M (t) PM (t)(1 − t)

Furthermore, if a ∈ / m2 , then

R −1 PR . M (t) = PM (t)(1 − t)

Proof. Note that (2.1.2.2) gives

R (t) = (1 − t)−1 . PR

The change of rings spectral sequence

R R TorR p (M, Torq (R, k)) ⇒ Torp+q (M, k)

(2.2.10.1)

22 R R gives an inequality PR M (t) PM (t) PR (t)

Assume now a ∈ / m2 . Since ν is large by Corollary 2.2.9, we can use [38, 1.⇒2.] to conclude that the following equality holds:

M R PM k (t) = Pk (t) · PR (t) .

We proceed now towards proving the main theorem. 2.2.11. Construction. Let Rhv, wi be the Tate resolution of R over R with v, w variables in homological degree 1, respectively 2, and ∂(v) = a, ∂(w) = bv. Note that it is a minimal free resolution of R. We have thus a surjective quasi-isomorphism of DG Γ algebras, extending the projection ν : R → R.

ϕ : Rhv, wi → R

Let s1 , . . . , se ∈ R be a sequence of elements such that their images s1 , . . . , se ∈ R form a minimal generating set for the maximal ideal of R. Let Rhyi, with variables y = y1 , y2 , . . . , be a minimal Tate resolution of k over R, with variables y = y1 , y2 , . . . such that ∂(yi ) = si for i = 1, . . . , e. By 2.2.1.1, the map ϕ extends to a surjective quasi-isomorphism

ϕ : Rhv, w, xi → Rhyi

(2.2.11.1)

such that ϕ(xi ) = yi for each i and ϕ6q : Rhv, w, x6q i → Rhy 6q i is a surjective quasi-isomorphism for each q. Note that ∂(xi ) = si for i = 1, . . . , e. In particular, the complex Rhv, w, xi is acylic and hence it is a free resolution of k

23 over R. This resolution may not be minimal. Proof of Theorem 2.2.3. With the notation set in Construction 2.2.11, we have TorR (k, k) = H(khv, w, xi), where khv, w, xi = Rhv, w, xi ⊗R k. Since the resolution Rhyi is minimal, TorR (k, k) = khyi, and the differential of khyi is zero. Let k(y) denote the vector space of indecomposables of the complex khyi, with zero differential. We have then π∗ (R) = ky. Also, let k(v, w) denote the indecomposables of the complex khv, wi with zero differential (induced from the minimal complex Rhv, wi), and let k(v, w, x) denote the indecomposables of khv, w, xi, with the induced differential. We have then an exact sequence of complexes:

0 → k(v, w) → k(v, w, x) → k(y) → 0

(2.2.11.2)

where the rightmost map is induced by ϕ. This sequence yields a long exact sequence in homology: di+1

ϕ ei

d

i . . . −−→ Hi (k(v, w)) → Hi (k(v, w, x)) − → Hi (k(y)) − → Hi−1 (k(v, w)) → . . .

(2.2.11.3) Note that k(v, w) has no elements in degrees higher than 2. The exact sequence (2.2.11.3) gives thus that ϕ ei is an isomorphism for all i ≥ 3. Set G = Rhv, w, xi. Claim. Set G = Rhv, w, xi. Then B2j (G) ⊆ mG2j + C(G)2j for all j. Indeed, if j > 1, the statement follows from the fact that ϕ e2j is an isomorphism, as established above. It suffices thus to assume j = 1. In this case, let x ∈ B2 (G). Since H2 (G) = 0, we have that B2 (G) = Z2 (G), hence ∂ G (x) = 0. We have x = P bw + bi xli + z, where z ∈ C(G)2 . Since ϕ(w) = 0 we conclude ϕ(x) = bi yli + ϕ(z). Note that ϕ(z) ∈ C(Rhyi). Since ϕ is a homomorphism of complexes, we know that

24 ϕ(x) is a cycle, thus a boundary of Rhyi. Since Rhyi is minimal, we know that bi ∈ n, hence bi ∈ m. Since any element of n is a boundary of Rhyi, note we can write ϕ(x) = ∂(c) + g where c ∈ C(Rhyi)3 and g ∈ C(Rhyi)2 . Regarding ϕ(x), c and g as elements of Rhy 62 i, it follows that [ϕ(x)] = [g] in this algebra. Note that any element g ∈ C(Rhyi)2 satisfies g (e+1) = 0 (a DG Γ-algebra generated by variables of degree 1 is nilpotent.) In consequence we have that [ϕ(x(e+1) )] = [0] in Rhy 62 i. Since the induced map ϕ62 : Rhv, w, x62 i → Rhy 62 i is a quasi-isomorphism, it follows that x(e+1) is the boundary of an element in Rhv, w, x62 i. We will show next that b ∈ m. Assume b ∈ / m. We may assume b = 1. The formula for the divided powers of a sum gives that w(e+1) appears with a coefficient of 1 in the expression of x(e+1) in terms of the basis elements of G2(e+1) . This contradicts the fact that x(e+1) is the boundary of an element in Rhv, w, x62 i. We conclude thus that b ∈ m as well. This finishes the proof of the claim. In [30, 3.2.1, 3.2.3] it is shown that the conclusion of the claim implies that there exists a homomorphism of DG Γ algebras G → X, where X is a minimal Tate resolution of k over R, inducing a quasi-isomorphism between the modules of indecomposables. Note that the module of indecomposables of X is π(R). Thus Hi (k(v, w, x)) ' π(R). Since the map π(ν) is naturally induced by ν and can be computed from any choice of Tate resolutions for R and R, we conclude that the map ϕ e in (2.2.11.3) can be identified with π(ν). The conclusion of the claim also implies that the differential of the complex k(v, w, x) is zero in odd degrees. An analysis of the connecting homomorphism di in the exact sequence (2.2.11.3) shows that di = 0 for all odd i. Note that k(v, w) has no elements in degrees higher than 2. The exact sequence (2.2.11.3) gives thus that πi (ν) is an isomorphism for all i ≥ 3 and there is an exact

25 sequence π2 (ν)

π1 (ν)

0 → kw → π2 (R) −−−→ π2 (R) → kv → π1 (R) −−−→ π1 (R) → 0

(2.2.11.4)

If we write R as R = Q/I where Q is a regular local ring and I is an ideal contained in q2 , where q denotes the maximal ideal of Q, then R = Q/(a0 , I), where a0 is the R preimage of a in Q. Since π1 (R) = TorR 1 (k, k) and π1 (R) = Tor1 (k, k), the map

π1 (ν) can be identified with the canonical projection q/q2 → q/(q2 , a0 ), and is an isomorphism if and only if a0 ∈ q2 , that is, a ∈ m2 . ∼ =

→ π1 (R) Thus, if a ∈ m2 , then (2.2.11.4) splits into an isomorphism π1 (ν) : π1 (R) − and an exact sequence

0 → kw → π2 (R) → π2 (R) → kv → 0

If a ∈ / m2 , since kv is a one-dimensional vector space and the map π1 (ν) is not injective, it follows that (2.2.11.4) splits into two exact sequences of k-vector spaces: π2 (ν)

0 → kw → π2 (R) −−−→ π2 (R) → 0 π1 (ν)

0 → kv → π1 (R) −−−→ π1 (R) → 0

2.3

Betti numbers

R A formula relating PR / m2 . M (t) and PM (t) was already given in 2.2.10, in the case a ∈

In this section we give a formula in the case a ∈ m Ann M . Theorem 2.3.1. Let (R, m, k) be a local ring and a ∈ R an exact zero divisor.

26 Set R = R/aR and let M be a finitely generated R-module with aM = 0. If a ∈ m Ann M , then R PR M (t) = PM (t)

Remark 2.3.2. Lescot [36] gives the following inequality:

R R R PR M (t) Pk (t) PM (t) Pk (t)

(2.3.2.1)

R R Note that, even if we know PR k (t) = Pk (t), we cannot deduce an inequality PM (t)

PR M (t) solely from (2.3.2.1) 2.3.3. Construction. Set J = Ann M and let a be such that a ∈ m Ann M . Using the notation set in Construction 2.2.11, let

a = s 1 a1 + · · · + s e ae

with ai ∈ J

such that s1 , . . . , se form a minimal generating set for m. Since ∂(xi ) = si for i = 1, . . . , e we have

∂(v − a1 x1 − · · · − ae xe ) = 0

and hence there exists an element T ∈ Rhv, w, xi such that

∂(T ) = v − a1 x1 − · · · − ae xe

Set S = R/J and form the DG Γ-algebras:

Shyi = Rhyi ⊗R S

and Shv, w, xi = Rhv, w, xi ⊗R S

27 In particular: ∂(v) = 0 and ∂(T ) = v

in Shv, w, xi

In general, if Ahti is a DG Γ extension of a ring A, and N is an A-module, we let N hti denote the DG-module Ahti ⊗A N Let F denote the DG-module M hv, w, xi. Since ∂(v) = 0 and v is an indeterminate of degree 1, one has an exact sequence of complexes:

0 → vF ,→ F → Σ−1 (vF ) → 0

(2.3.3.1)

where the rightmost map is given by multiplication by v. Recall that there is a surjective quasi-isomorphism ϕ : Rhv, w, xi → Rhyi with ϕ(w) = ϕ(v) = 0. The induced surjective morphism ϕ ⊗R M : F → M hyi factors through vF , inducing a surjective morphism ϕ : F/vF → M hyi. Let G denote the kernel of ϕ, so that there is an exact sequence of complexes ϕ

0 → G ,→ F/vF − → M hyi → 0

(2.3.3.2)

We can now give the proof of the main theorem. We use the notation in the Construction above. Proof of Theorem 2.3.1. We first establish the coefficient-wise inequality

R PR M (t) PM (t)

(2.3.3.3)

This inequality is proved in a sequence of three steps. Step 1. We prove that the inclusion vF ,→ F induces the zero map in homology. More generally, we prove by induction on n the following statement: If x ∈ F is an element

28 of degree n such that ∂(vx) = 0, then the element vxT (i) is a boundary for every i ≥ 0. The desired statement then follows from the case i = 0. Note that the relation ∂(vx) = 0 implies v∂(x) = 0 and thus ∂(x) ∈ vF . If ∂(x) = 0, then one has ∂(xT (i+1) ) = vxT (i) , and this settles the cases n = 0 and n = 1. Let n ≥ 1 and assume the statement to be proved when x has degree at most n. Consider then x to have degree n + 1. We have ∂(x) = vz for some z ∈ F of degree n − 1. Since ∂(vz) = 0, the induction hypothesis gives that vzT (i+1) is a boundary. It follows that vxT (i) is a boundary as well, since

∂(xT (i+1) ) = vzT (i+1) + vxT (i) Step 2. We construct an isomorphism of complexes F/vF ∼ = Σ−2 G. Let w denote the image of the variable w in the DG algebra Shv, w, xi/vShv, w, xi. Since ∂(w) = v in Shv, w, xi, one has ∂(w) = 0. Note that every element u ∈ F/vF can be uniquely expressed as u=

X

w(i) ui

i≥0

where ui are images of elements in M hxi. (In the sum, at most finitely many elements are nonzero, due to degree considerents). Noting that ϕ(u) = ϕ(u0 ), we conclude that u ∈ G if and only if u0 = 0. We define then a map ψ : F/vF → Σ−2 F/vF by

ψ(

X i≥0

w(i) ui ) =

X i≥0

w(i+1) ui =

X

w(j) uj−1

j≥1

Since ∂(w) = 0, note that ψ is a morphism of complexes. Also, ψ is injective and its image is Σ−2 G. Step 3. Recall from Construction 2.2.11 that Rhv, w, xi is a free resolution of k over

29 R, and Rhyi is a minimal free resolution of k over R. This justifies the first and the last equality below:

dimk (TorR i (M, k)) = dimk (F ) = dimk (Hi+1 (vF )) − dimk (Hi−1 (vF )) = dimk (Hi (F/vF ) − dimk (Hi−2 (F/vF )) = dimk (Hi (F/vF ) − dimk (Hi (G)) ≤ dimk (Hi (M hyi)) = dimk (TorR i (M, k))

The second equality follows from a dimension count in (2.3.3.1), using the fact that the inclusion vF ,→ F induces the zero map in homology. The third equality follows from the isomorphism of complexes F/vF ∼ = Σ−1 (vF ), as given by the exact sequence (2.3.3.1). The fourth equality follows from the isomorphism F/vF ∼ = Σ−2 G, and the last inequality follows from a dimension count in the long exact sequence induced by (2.3.3.2). R Using (2.3.2.1) and the equality PR k (t) = Pk (t) given by Corollary 2.2.5, we

conclude R R R R R PR M (t) Pk (t) = PM (t) Pk (t) PM (t) Pk (t)

In consequence equality holds in (2.3.2.1), thus:

R R R PR M (t) Pk (t) = PM (t) Pk (t)

R Using again (2.3.3.3), we conclude that PR M (t) = PM (t).

30

2.4

Poincar´ e series of graded modules

The main results of this section are Theorem 2.4.2 and a graded version of Corollary 2.2.10 (Theorem 2.4.2), which permits an analysis of the Koszul property modulo an exact zero-divisor. We begin by fixing some standard terminology for graded rings and modules. 2.4.1. Graded Poincaré series. Let k be a field and let A = ⊕n≥0 An be a standard graded k-algebra (in the sense of [32, 1.8]). Recall that each graded module N over A has a graded free resolution and any such resolution induces a natural grading on

TorA i (N, k) =

M

TorA i (N, k)j .

j

One defines the graded Betti numbers of N over A to be

A βi,j (N ) = rankk TorA i (N, k)j

and the graded Poincaré series of N over A is defined to be

PA N (s, t) =

X

A βi,j (N ) si tj

i,j

Thus PA N (s, t) is a formal power series in t with coefficients in the Laurent polynomial ring in s: ±1 PA [[t]] N (s, t) ∈ Z s As in [41, §2], we say that a finite graded module N over a standard graded k-algebra A has a linear resolution if TorA i (N, k)j = 0 for j 6= i; equivalently, if the graded Poincaré series PA N (s, t) can be written as a formal power series in the product

31 st. Theorem 2.4.2. Let A be a standard graded k-algebra and a, b be elements in A1 forming an exact pair of zero divisors on A. For every finite positively graded A/aAmodule N , one then has A/aA

PA N (s, t) = PN

(s, t) ·

1 1 − st

(2.4.2.1)

Proof. The proof of this theorem is a graded version of the previous proof. A bookkeeping exercise shows that the proof in [38, 1.1, (3)⇒(2)] translates smoothly to the graded case. Namely, that argument uses spectral sequences that preserve grading and lead to the following conclusion: A/aA

PA N (s, t) = PN

A/aA

(s, t) · PA A/aA (s, t) = PN

(s, t) ·

1 . 1 − st

We discuss next the Koszul property for rings and modules. We refer the reader to [32] and [41] for detailed studies of the Koszul property. Throughout we adopt the following notation for the associated graded objects over a local ring (R, m, k): let Rg denote the associated graded ring and let M g denote the associated graded module of an R-module M ; that is,

Rg =

M n≥0

mn /mn+1

and M g =

M

mn M/mn+1 M .

n≥0

2.4.3. Koszul modules. As in [32], we say that a finite module M over a local ring (R, m, k) is Koszul if for a minimal free resolution F of M the complex

linR (F ) = · · · −→ Fn+1 g (−n − 1) −→ Fn g (−n) −→ · · · −→ F0 g −→ M g −→ 0

32 with differentials induced from F is acyclic. A local ring R is said to be Koszul if its residue field k is Koszul as a module over R. From [41, 2.3] and [32, 1.5] the following statements are equivalent for a finite module M over a local ring R: 1. M is a Koszul R-module 2. M g has a linear resolution over Rg Rg

3. Tori (M g , k)j = 0 for j 6= i Rg

4. PM g (s, t) can be written as a formal power series in the product st. Further, as proved in [32, Prop. 1.8], the Poincaré series of a Koszul module M over a local ring R satisfies Rg

−1 PR M (t) = PM g (t) = HM (−t) HR (−t) .

(2.4.3.1)

A standard graded k-algebra A is said to be a Koszul algebra if k admits a linear free resolution as a graded module over A. It follows that a local ring R is Koszul if and only if Rg is Koszul (as a standard graded k-algebra). Consequently, a local ring Rg

is Koszul if and only if Pkg (1, t) = HR (−t)−1, as shown in [26, Thm 1]. 2.4.4. Graded Koszul modules. A graded module N over a standard graded k-algebra A is said to be Koszul if for a minimal graded free resolution F of N , letting m denote the irrelevant maximal ideal of A, the complex of Am -modules linA (Fm ) is acyclic. If a graded A-module N has a linear resolution, then it is Koszul, but the converse fails in general; see [32, 1.9]. Corollary 2.4.5. Let A be a standard graded k-algebra and a, b be elements in A1 forming an exact pair of zero divisors on A. A finite positively graded A/aA-module

33 N has a linear resolution over R if and only if it has a linear resolution over R/aR. In particular, the ring A is Koszul if and only if the ring A/aA is Koszul. Proof. Set A = A/aA. As a consequence of Theorem 2.4.2, PA N (s, t) is a power series in the product st precisely when PA N (s, t) is a power series in the product st. In this way, it becomes clear that M has a linear resolution over A if and only if it has one over A. It follows immediately from the case N = k, that the ring A is Koszul if and only if the ring A/aA is Koszul.

34

Chapter 3 Exact zero divisors over Artinian Gorenstein rings Introduction This chapter is organized as follows. In section 1, we introduce the notion of a balanced Hilbert series and show that the Hilbert series of R and R/aR are closely linked, when a is an exact zero divisor. In section 2, we establish our main result, Theorem 3.2.3. The rings considered are proved to be Koszul homomorphic images of a codimension two complete intersection ring under a Golod homomorphism. Results of Herzog and Iyengar [32] then yield that every finite module over R has a Koszul syzygy module. (See 2.4.3 for the definition of a Koszul module.) In the last section of this chapter, we study Gorenstein rings and deduce Theorem 1.

35

3.1

Short local rings with balanced Hilbert series

In this section we study properties of exact pairs of zero divisors over local rings R satisfying m4 = 0 and HR (−1) = 0. Theorem 3.1.5 establishes that the initial forms a∗ , b∗ of an exact pair of zero divisors a, b in R remain an exact pair of zero divisors in Rg , yielding a numerical relation between the Hilbert series of R and R/aR. We consider both local and graded rings, hence we proceed to clarify terminology. If A is a standard graded k-algebra and N = ⊕n≥s Nn is a finite A-module, then the Hilbert series of N is the formal power series

HN (t) =

X

rankk (Nn )tn .

n≥s

If A is artinian, then A is in particular a local ring and this notion of Hilbert series coincides with the one defined for a local ring, whenever the module is generated in degree 0. In this setting, we will not distinguish between the local notion and the graded notion. We say that a module N has balanced Hilbert series if HN (−1) = 0. Lemma 3.1.1. Let A be a standard graded k-algebra and a ∈ R1 . If N is a finite graded A-module such that there exists an isomorphism of graded A-modules N/aN (−1) ∼ = (0 :N a), then

HN (t) = (1 + t) HN/aN (t).

(3.1.1.1)

In particular, if a, b ∈ A1 form an exact pair of zero divisors then (3.1.1.1) holds with N = A, hence A has balanced Hilbert series.

36 Proof. The additivity property of the Hilbert series in the exact sequence a

0 → (0 :N a)(−1) → N (−1) → − N → N/aN → 0

gives an equality

t H(0:N a) (t) − t HN (t) + HN (t) − HN/aN (t) = 0 . From the isomorphism N/aN (−1) ∼ = (0 :N a) of graded R-modules we then have an equality (1 − t) HN (t) = (1 − t2 ) HN/aN (t) which establishes (3.1.1.1) If a, b ∈ A1 form an exact pair of zero divisors, the complex f defined in 2.1.2.2is exact and gives an isomorphism A/aA(−1) ∼ = (0 :A a) of graded A-modules yielding the desired conclusion. Let (R, m, k) be a local ring. When M is a finite R-module we let µ(M ) denote the minimal number of generators of M over R; thus, one has µ(M ) = rankk M/mM . Also, λ(M ) denotes the length of M . When mM = 0, one has µ(M ) = λ(M ). We collect below several basic facts used throughout. Remark 3.1.2. Let (R, m, k) be a local ring and a ∈ m. The following then hold: 1. If a ∈ / m2 , then aR ∩ m2 = am. 2. If a 6= 0, then λ(aR) = λ(am2 ) + µ(am) + 1. 3. λ(0 :R a) = λ(R) − λ(aR). 4. If m4 = 0, then λ(R) = 1 + µ(m) + µ(m2 ) + µ(m3 ).

37 Indeed, to prove (1), let x ∈ R such that ax ∈ m2 . Since a is not contained in m2, we conclude that x is not an unit, hence ax ∈ am. Formula (2) is given by a length count in the exact sequences:

0 → am2 → am → am/am2 → 0 0 → am → aR → aR/am → 0 .

Formula (3) is given by a length count in the exact sequence

0 → (0 :R a) → R → aR → 0 ,

and (4) by a length count on the quotients of the m-adic filtration:

0 = m4 ⊂ m3 ⊂ m2 ⊂ m ⊂ R .

Proposition 3.1.3. Let (R, m, k) be a local ring with m4 = 0 and balanced Hilbert series. For a, b in m r m2 , the following statements are equivalent: (1) a, b is an exact pair of zero-divisors (2) (0 : a) = bR (3) ab = 0, am2 = m3 = bm2 and µ(am) = µ(m) − 1 = µ(bm). Proof. It is clear that (1) implies (2). For the rest it suffices to show that (2) and (3) are equivalent; indeed, if this is known, we see that (2) implies (0 : b) = aR, since (3) is symmetric with respect to a and b. We may assume ab = 0 as this holds under any of the conditions (1) to (3).

38 Set e = µ(m) and s = µ(m3 ). Since m4 = 0, one gets µ(am2 ) = λ(am2 ) and µ(bm2 ) = λ(bm2 ). Remark 3.1.2(2) gives

λ(aR) = µ(am2 ) + µ(am) + 1 .

(3.1.3.1)

Since b 6∈ m2 , we may choose b, x2 , x3 , . . . , xe to be a minimal system of generators for m. As ab = 0, we see that am is generated by ax2 , . . . , axe , hence

µ(am) ≤ e − 1 .

(3.1.3.2)

From the inclusion am2 ⊆ m3 we obtain the obvious inequality

µ(am2 ) ≤ s

(3.1.3.3)

with equality if and only if am2 = m3 . Using (3.1.3.2) and (3.1.3.3) in (3.1.3.1) we thus have λ(aR) ≤ e + s

(3.1.3.4)

with equality if and only if µ(am) = e − 1 and am2 = m3 . Similarly, we obtain

λ(bR) ≤ e + s

(3.1.3.5)

with equality if and only if µ(bm) = e − 1 and bm2 = m3 . The hypothesis HR (−1) = 0 implies λ(R) = 2(e + s) and Remark 3.1.2(3) gives

λ(0 : a) = 2(e + s) − λ(aR) .

39 Thus the inequality (3.1.3.4) yields

λ(0 : a) ≥ e + s

(3.1.3.6)

with equality if and only if µ(am) = e − 1 and am2 = m3 . Since ab = 0, we have bR ⊆ (0 : a) and the inequalities above yield

λ(bR) ≤ e + s ≤ λ(0 : a).

We have thus bR = (0 : a) if and only if equalities hold in (3.1.3.5) and (3.1.3.6), that is, if and only if condition (3) is satisfied. Lemma 3.1.4. Let (R, m, k) be a local ring satisfying m4 = 0 and µ(m) = e ≥ 3. (1) If R has balanced Hilbert series and an exact zero divisor lies in m r m2 then so do its complementary divisors. (2) If µ(m3 ) + 2 ≤ e, then a non-zero exact zero divisor is not contained in m2 . Proof. Let f = µ(m2 ) and g = µ(m3 ). (1) As m4 = 0, we have HR (t) = 1+et+f t2 +gt3 . If HR (−1) = 0 then g = f −e+1. Now let a ∈ m r m2 be an exact zero divisor and b its complementary divisor. Assume that b is in m2 . One then has

aR = (0 :R b) ⊇ (0 :R m2 ) ⊇ m2 .

Hence m2 = am by Remark 3.1.2 (1) and

f = µ(m2 ) = µ(am) ≤ µ(m) = e.

40 We have thus g = f − e + 1 ≤ 1. Since e ≥ 3, a contradiction is derived from (2). (2) Assume b is a non-zero exact zero divisor in m2 and let a be a complementary zero divisor. Then bm ⊆ m3 hence

λ(bR) = λ(bR/bm) + λ(bm) ≤ µ(bR) + λ(m3 ) = 1 + g.

On the other hand, am ⊆ m2 as a ∈ m, so

λ(aR) = λ(aR/am) + λ(am) ≤ µ(aR) + λ(m2 ) = 1 + µ(m2 ) + µ(m3 ) = 1 + f + g.

From Remark 3.1.2 (4) and (3), we have

1 + e + f + g = λ(R) = λ(aR) + λ(bR) ≤ 1 + f + g + 1 + g

implying e ≤ g + 1, a contradiction. We use the notation introduced in Section 1: Rg and M g denote the associated graded ring and module respectively. Given an ideal I of R, let I ∗ denote the ideal of Rg generated by the initial forms of elements in I. Also, let r∗ denote the initial form of an element r in R. Theorem 3.1.5. Let (R, m, k) be a local ring with m4 = 0 and balanced Hilbert series. If a, b ∈ m r m2 form an exact pair of zero divisors then the following hold: (1) their initial forms a∗ , b∗ in Rg form an exact pair of zero divisors; (2) the graded rings (R/aR)g and Rg /a∗ Rg are naturally isomorphic; (3) HRg /a∗ Rg (t) = HR/aR (t) = HR (t)(1 + t)−1 .

41 Proof. (1) Note that (Rg , mg ) is a local ring, where mg denotes the irrelevant maximal g ideal R>1 of Rg . One has

µ(am) = λ(am/am2 ) = µ(a∗ mg ) and µ(m) = λ(m/m2 ) = µ(mg ) .

Further, the property am2 = m3 = bm2 given by Proposition 3.1.3 implies

a∗ (mg )2 = (mg )3 = b∗ (mg )2 .

Proposition 3.1.3 shows that a∗ , b∗ form an exact pair of zero divisors in Rg . (2) The canonical homomorphism R → R/aR induces a surjective homomorphism of graded k-algebras ϕ : Rg → (R/aR)g . From [46, (1.1)], ker ϕ = a∗ Rg if and only if

mi ∩ aR = ami−1

for all i ≥ 1 .

(3.1.5.1)

The cases i = 1 and i ≥ 4 are obvious and case i = 2 is given by Remark 3.1.2(1). Finally, i = 3 follows from Proposition 3.1.3(3) and we conclude that (3.1.5.1) holds. (3) Note that HRg (t) = HR (t) and HRg /a∗ Rg (t) = H(R/aR)g (t) = HR/aR (t). The statement follows then from (1) and Lemma 3.1.1.

3.2

Modules over short local rings

In this section we study the Koszul property of finite modules over local artinian rings (R, m, k) with m4 = 0 and balanced Hilbert series. We establish our main result, Theorem 3.2.3, whose primary case of interest (over Gorenstein rings) is discussed in section 4. An important structural result is Proposition 3.2.9 yielding that, in the presence of special elements a, b, c ∈ R, the

42 rings considered are homomorphic images of a codimension 2 complete intersection by means of a Golod homomorphism. This allows to apply results of Herzog and Iyengar [32, §5] on the Koszul property. Proposition 3.2.1. Let (R, m, k) be a local ring with m4 = 0 and balanced Hilbert series. If a, b in m r m2 form an exact pair of zero divisors and M is an R/aR-module, then M is Koszul over R if and only if it is Koszul over R/aR. In particular, the ring R is Koszul if and only if the ring R/aR has this property. Proof. Recall from 2.4.3 that a module is Koszul over a local ring if and only if its associated graded module has a linear resolution over the associated graded ring. From Theorem 3.1.5(1), the initial forms a∗ , b∗ form an exact pair of zero divisors in Rg . As M g is a positively graded Rg /a∗ Rg -module, Corollary 2.4.5 shows that M g has a linear resolution over Rg precisely when M g has a linear resolution over Rg /a∗ Rg . This is the desired conclusion because the graded rings Rg /a∗ Rg and (R/aR)g are naturally isomorphic by Theorem 3.1.5(2). Recall that the nth syzygy module of a finite R-module M is the cokernel of the nth differential in a minimal free resolution of M over R. It is defined uniquely up to R isomorphism and we denote it ΩR n (M ). If an R-module N is isomorphic to Ωn (M ) for

some n ≥ 0, then we say that N is a syzygy module of M . 3.2.2. Koszul syzygy modules. If (R, m, k) is a zero-dimensional local ring and M a finite R-module admitting a Koszul syzygy module then

HR (−t) PR M (t) ∈ Z[t].

This follows immediately from [32, Prop. 1.8].

43 In this way, if R is a zero-dimensional local ring over which every finite R-module M admits a Koszul syzygy module, then for each finite R-module M there exits a polynomial pM (t) ∈ Z[t] satisfying PR M (t) = pM (t)/HR (−t).

In general an uniform bound for the degree of pM (t) can not be derived. Indeed, if R is a self-injective non-regular Koszul ring, then for every n ≥ 0 there exists a R −1 non-zero finite R-module M satisfying k ∼ one then = ΩR n (M ). Since Pk (t) = HR (−t)

has n R PR M (t) = t Pk (t) + qM (t) =

tn pM (t) + qM (t) = HR (−t) HR (−t)

for some (uniquely determined) polynomials qM (t) and pM (t) of degrees n − 1 and n − 1 + deg HR (t) respectively. Our main result is as follows. Theorem 3.2.3. Let (R, m, k) be a local ring with HR (t) = 1 + et + et2 + t3 and set s = rankk (0 :R m). Assume there exists a non-zero exact zero divisor a ∈ m. The following statements are then equivalent: (1) R is Koszul. (2) e ≥ s + 2. When they hold every finite R-module M has a syzygy module that is Koszul. The proof of Theorem 3.2.3 is deduced from Theorem 3.2.11. Corollary 3.2.4. Let (R, m, k) be a local ring with HR (t) = 1 + et + et2 + t3 and e ≥ 3. Assume there exists a non-zero exact zero divisor a ∈ m and Rg is quadratic.

44 The ring R then is Koszul and every finite R-module M has a syzygy module that is Koszul. Proof. Let b be a complementary divisor for a, then a, b form an exact pair of zero divisors in m r m2 by Lemma 3.1.4. Theorem 3.1.5(1) then establishes that the initial forms a∗ , b∗ of a, b form an exact pair of zero divisors in Rg . Further, Rg /a∗ Rg is quadratic and HRg /a∗ Rg (t) = 1 + (e − 1)t + t2 by Theorem 3.1.5(3). By [24, 2.12], it follows that rankk (0 :Rg mg ) ≤ e − 2 Applying Theorem 3.2.3 to the local ring Rg , we conclude that Rg is Koszul, hence R is Koszul as well. Thus R satisfies conditions (1)-(2) of the same theorem, concluding the proof. Remark 3.2.5. Let R be an artinian graded k-algebra with irrelevant maximal ideal m. In particular, (R, m, k) is a local ring. In [24], Conca, Rossi and Valla consider standard graded k-algebras R with HR (t) = 1 + et + et2 + t3 , where e ≥ 3, that contain elements l, m ∈ R1 with

lm = 0

and

rankk (lR1 ) = rankk (mR1 ) = e − 1 .

(3.2.5.1)

When R is quadratic or Gorenstein, Proposition 3.1.3 and the proof of [24, 2.13] show that a pair l, m ∈ R1 satisfies (3.2.5.1) if and only if it is an exact pair of zero divisors. The Gorenstein case also follows from Proposition 3.3.1 in Section 4. We introduce terminology to handle the technical ingredients of the proof. 3.2.6. Conca generators. Following [8], we say that an element c ∈ m is a Conca

45 generator modulo aR if it satisfies

m2 + aR = cm + aR,

c∈ / aR and c2 ∈ aR.

Let c be the image of c in R. These conditions imply m3 = 0 and c ∈ / m2 = cm by Nakayama’s Lemma. Further, they are equivalent to

m2 ⊆ cm + am,

c∈ / aR and c2 ∈ am

whenever a is part of an exact pair of zero divisors, as given by (3.1.5.1). In the terminology of [8], the image in R/aR of a Conca generator modulo aR is exactly a Conca generator of the maximal ideal m/aR. 3.2.7. Golod homomorphisms. The notion of Golod homomorphism was introduced by Levin [37]. An epimorphism ψ : (Q, q, k) (R, m, k) is said to be Golod if the following equality holds −1

Q Q PR k (t) = Pk (t)(1 − t(PR (t) − 1)) .

(3.2.7.1)

In the proof of the next proposition we use a result of Valabrega and Valla: 3.2.8. Initial forms and regularity. To establish the regularity of a sequence in a local ring Q it suffices to establish regularity of the corresponding sequence of initial forms in Qg ; see [46, 2.7]. As R is artinian, Cohen’s Structure Theorem yields a regular local ring (P, p, k) and a surjective local homomorphism

π : (P, p, k) (R, m, k)

46 such that ker π ⊆ p2 . Proposition 3.2.9. Let (R, m, k) be a local ring with m4 = 0 and balanced Hilbert series. Let a, b, c be elements in m r m2 such that a, b form an exact pair of zero divisors and c is a Conca generator modulo aR. The ring R then is Koszul, and there exist u, v ∈ p r p2 and w ∈ p2 such that uv, w is a P -regular sequence contained in ker π and such that the homomorphism

Q = P/(uv, w) R

induced by π is Golod. Proof. Let (R, m) denotes the local ring (R/aR, m/aR). For each x ∈ R we let x denote the image of x in R. The ring R is Koszul by [8, 1.1], so Proposition 3.2.1 shows that R is Koszul, and from (2.4.3.1) we obtain

−1 PR k (t) = HR (−t) .

(3.2.9.1)

The rest of the proof is conducted in three steps. The first one is to show that π factors through a map ψ : Q R where Q is a complete intersection of the form P/(uv, w) for some regular sequence uv, w in P such that π(u) = a. We let (P , p) denote the regular local ring (P/uP, p/uP ) and for every x ∈ P we denote x the image of x in P . In a second step we remark that the induced map π : P R is a Cohen presentation for R with ker π ⊆ p2 and maps a minimal generator for p to a Conca generator for m. We then use [8, 1.4] to conclude that the map ψ : Q R induced by ψ on Q = Q/˜ uQ is Golod, where u˜ denotes the image of u in Q.

47 The final step consists of establishing the equality (3.2.7.1) and therefore concluding that ψ itself is a Golod homomorphism. Step 1. We show there exist elements u, v in p r p2 and w in p2 such that π(u) = a, π(v) = b and uv, w forms a P -regular sequence. The map π then factors through a homomorphism

ψ : (Q, q, k) (R, m, k) where Q = P/(uv, w).

Indeed, as c is a Conca generator modulo aR, there exists d ∈ m such that

c2 = ad and m2 + (a) = cm + (a).

(3.2.9.2)

One then has c ∈ / m2 so a, c can be extended to a minimal generating set for m. There exists thus a minimal system of generators x1 , . . . , xe for p satisfying π(x1 ) = a and π(x2 ) = c. Set Xi = xi ∗ in P g and note that P g is the polynomial ring over k on X1 , . . . , Xe , since P is regular. Pick y ∈ p such that π(y) = d and set

u = x1

and w = x22 − yx1 .

To carry out the first step we consider three cases. Case 1. rankk {a∗ , b∗ } = 1. We have thus a∗ Rg = b∗ Rg . We may then assume b = a + f for some f ∈ m2 , since exact pairs of zero divisors are preserved up to multiplication by a unit in R. Choose

48 z ∈ p2 so that π(z) = f ; the map π then factors through the map P ψ −−→ R. 2 (x1 (x1 + z), x2 − yx1 )

Q=

(3.2.9.3)

Set v = x1 + z. We claim that the sequence uv, w is P -regular. In P g one has

∗

(uv) = X1

   X2 2 − y ∗ X 1 , ∗ and w =   X 2,

2

2

if y ∈ p r p2

.

if y ∈ p2

Since both sequences

X1 2 , X2 2

and X1 2 , X2 2 − y ∗ X1

are k[X1 , . . . , Xe ]-regular, we conclude from 3.2.8 that uv, w is P -regular. Case 2. rankk {a∗ , b∗ , c∗ } = 3. As a, b, c can be extended to a minimal generating set for m, then we may assume π(x3 ) = b. The map π then factors through the homomorphism

Q=

P ψ −−→ R. 2 (x1 x3 , x2 − yx1 )

(3.2.9.4)

Set v = x3 . We claim that the sequence uv, w is P -regular. In P g one has

∗

(uv) = X1 X3

   X 2 2 − y ∗ X1 , ∗ and w =   X 2, 2

if y ∈ p r p2 if y ∈ p2

Since both sequences

X1 X3 , X2 2

and X1 X3 , X2 2 − y ∗ X1

.

49 are k[X1 , . . . , Xe ]-regular, we conclude from 3.2.8 that uv, w is P -regular. Case 3. rankk {a∗ , b∗ , c∗ } = 2 = rankk {a∗ , b∗ }. In other terms, c = b + g in R/aR for some g ∈ m2 . Since m3 = am2 = 0, we have 2

b = c2 = 0 and m2 = m c = m b. Thus we may replace c with b and then (3.2.9.2) yields b2 = ad. If d ∈ bR, then b2 = ad = 0 which implies b ∈ (0 :R b) = aR. As b ∈ / m2 , it would then fall into Case 1. It remains to treat the case when d ∈ / bR. If d ∈ / bR, the map π then factors through the homomorphism

Q=

P ψ −−→ R. 2 (x1 x2 , x2 − yx1 )

(3.2.9.5)

Set v = x2 . The sequence uv, w is P -regular: because P is a UFD, x1 , x2 are irreducible elements and x2 - y. Step 2. We observe that the map ψ : Q R induced by ψ : Q R on Q = Q/˜ uQ is Golod. By choice, u = x1 is not in p2 , so the ring P = P/uP is regular. The induced map π : P R is a Cohen presentation with ker π ⊆ p2 such that the element x2 ∈ p r p2 maps to the Conca generator c of m. Note that there are canonical isomorphisms

Q=

Q ∼ P ∼ P = = 2 . u˜Q (u, w) x2 P

(3.2.9.6)

Under this identification the map π : P R factors through ψ : Q R, and it follows from [8, 1.4] that ψ is a Golod homomorphism. Step 3. We now prove that the homomorphism ψ : Q R is Golod.

50 We first note that, from Remark 2.1.4, the images u˜, v˜ of u, v in Q form an exact pair of zero divisors. Thus, as in Remark 2.1.2.2, Fu˜,˜v (Q) is a minimal free resolution of Q over Q. Since ψ(˜ u) = a and ψ(˜ v ) = b, one has an isomorphism of complexes Fu˜,˜v (Q) ⊗Q R ∼ = Fa,b (R),

which induces isomorphisms

TorQ ˜,˜ v (Q) ⊗Q R) = Hn (Fa,b (R)) = 0, for all n > 0. n (Q, R) = Hn (Fu

Therefore, PRQ (t) = PRQ (t).

(3.2.9.7)

Since P is a regular local ring, (3.2.9.6) and [45, Thm 6] yield equalities

PkQ (t) = (1 + t)e (1 − t2 )−2 ,

(3.2.9.8)

PkQ (t) = (1 + t)e−1 (1 − t2 )−1 ,

(3.2.9.9)

where e = µ(m) = µ(m) + 1. Now we can write the following sequence of equalities

PR k (t)=

PQ k (t) (1 − t(PRQ (t) − 1))(1 − t)

=

(1 + t)e−1 (1 − t(PRQ (t) − 1))(1 − t2 )(1 − t)

PkQ (t) (1 + t)e = = (1 − t(PRQ (t) − 1))(1 − t2 )2 1 − t(PRQ (t) − 1) The first equality above follows from Step 2 and Corollary 2.2.10, the rest from (3.2.9.9), (3.2.9.7) and (3.2.9.8) respectively. The resulting equality yields that the map ψ : Q R is Golod.

51 Remark 3.2.10. Let ϕ : (R, m, k) ,→ (R0 , m0 , k 0 ) be an inflation; that is, a flat homomorphism of local rings such that ϕ(m) = m0 . One then has HR (t) = HR0 (t). For every local ring (R, m, k) there exits a local ring (R0 , m0 , k 0 ) and an inflation R ,→ R0 where k 0 is an algebraically closed field k, see [18, AC IX.41]. 0

An inflation ϕ : (R, m, k) ,→ (R , m0 , k 0 ), where R = R/aR, m = m/aR and 0

(R , m0 , k 0 ) is a local ring, lifts to an inflation ϕ : (R, m, k) ,→ (R0 , m0 , k 0 ) satisfying 0 R0 /ϕ(a)R0 ∼ = R , see [18, AC IX.40].

Given a minimal free resolution F of an R-module M , the complex F ⊗R R0 is a minimal free resolution of the R0 -module M 0 = M ⊗R R0 . Thus one gets 0 0 ∼ R 0 ΩR n (M ) = Ωn (M ) ⊗R R

and

0

R PR M 0 (t) = PM (t).

Further, as noted in [8, 1.8], the extended module M 0 is Koszul over R0 precisely when M is Koszul over R. Theorem 3.2.11. Let (R, m, k) be a local ring with m4 = 0 and HR (−1) = 0. Assume that there exist an inflation (R, m, k) → (R0 , m0 , k 0 ) and elements a, b, c in 2

m0 r m0 such that a, b form an exact pair of zero divisors in R0 and c is a Conca generator modulo aR0 . The ring R then is Koszul and every finite R-module M has a syzygy module that −1 is Koszul. In particular, PR ∈ Z[t]. M (t) HR (−t)

Proof. In view of Remark 3.2.10 and 2.1.3, we can replace (R0 , m0 , k 0 ) with (R, m, k) and M ⊗R R0 with M . The ring R then satisfies the hypothesis of Proposition 3.2.9, and hence is Koszul. Proposition 3.2.9 and [32, 5.9] show that every finite R-module M has a syzygy −1 module that is Koszul over R; hence PR ∈ Z[t] from 3.2.2. M (t) HR (−t)

52 Proof of Theorem 3.2.3. Let b be a complementary divisor for a. From Lemma 3.1.4(2) a, b form an exact pair of zero divisors in m r m2 , since µ(m3 ) = 1 = 3 − 2 ≤ e − 2. Recall that R = R/aR is a local ring with maximal ideal m = m/aR and HR (t) = 1 + (e − 1)t + t2 as given by Theorem 3.1.5(3), then

rankk (m/m2 ) = e − 1 and rankk (m2 ) = 1.

Lemma 2.1.8 yields rankk (0 :R m) = s and [8, 4.1] shows that R is Koszul if and only if

rankk (0 :R m) ≤ rankk (m/m2 ) − 1

that is, s ≤ e − 2. Proposition 3.2.1 gives that R is Koszul if and only if R is Koszul, establishing thus the equivalence of (1) and (2). Assuming that (1) and (2) hold, then [8, Thm 4.1,(v) =⇒ (i)] and Remark 3.2.10 establish the existence of an inflation ϕ : (R, m, k) ,→ (R0 , m0 , k 0 ) such that m0 r m0 2 contains an element c that is a Conca generator modulo ϕ(a)R0 . Finally, from 2.1.3 we can apply Theorem 3.2.11 concluding the proof.

3.3

Short local Gorenstein rings

In this section we study exact pairs of zero divisors in local Gorenstein rings (R, m, k) with m4 = 0. We prove the Gorenstein version of our main result as presented in the introduction. Recall that R is Gorenstein if and only if R = R/aR is Gorenstein, as discussed in

53 Corollary 2.1. Proposition 3.3.1. Let (R, m, k) be a Gorenstein local ring satisfying m4 = 0 and µ(m) = e ≥ 3. Given 0 6= a ∈ m the following statements are equivalent: (1) (0 :R a) is a principal ideal; (2) a is an exact zero divisor; (3) a is in m r m2 and there exists b ∈ m r m2 such that the following equivalent conditions hold: (3.1) a, b form an exact pair of zero divisors; (3.2) ab = 0 and µ(am) = e − 1 = µ(bm) . If any of the conditions (1)-(3) holds, then HR (t) = 1 + et + et2 + t3 .

Proof. We first show that given a, b ∈ m r m2 , (3.1) implies HR (t) = 1 + et + et2 + t3 . Set (R, m) = (R/aR, m/aR). The hypothesis that a ∈ m r m2 implies µ(m) = e − 1. Since R is Gorenstein, Corollary 2.1 shows that R is Gorenstein. Note that m2 = (m2 , aR)/aR ∼ = m2 /aR ∩ m2 = m2 /am

(3.3.1.1)

where the equality aR ∩ m2 = am is given by Remark 3.1.2(1). Assume m2 = 0. Since R is Gorenstein, it follows that m = 0 or µ(m) = 1, hence e = 1 or e = 2. We may then assume m2 6= 0. If m3 = 0, then, since R is Gorenstein, m2 6= 0 and a ∈ m r m2 , it follows that m2 = am and thus m2 = 0, a contradiction.

54 We have thus m3 6= 0. Since R is Gorenstein with m4 = 0, we have m3 = (0 :R m) and µ(m3 ) = 1. Remark 3.1.2(4) gives

λ(R) = µ(m2 ) + e + 2.

(3.3.1.2)

The inclusion am2 ⊆ m3 implies am2 = m3 or am2 = 0. If am2 = 0, then am ⊆ (0 :R m). Since a ∈ m r m2 , we have am 6= 0, hence am = (0 :R m) = m3 . We have thus one of the two cases: (i) am = m3 and am2 = 0, hence µ(am) = 1 and µ(am2 ) = 0. (ii) am2 = m3 , hence µ(am2 ) = 1. In either case, m3 ⊆ am, hence m3 = 0 6= m2 . Since R is Gorenstein, we conclude that rankk (m2 ) = 1, hence rankk (m2 /am) = 1 by (3.3.1.1). A length count in the exact sequence 0 → am/m3 → m2 /m3 → m2 /am → 0 gives rankk (am/m3 ) = µ(m2 ) − 1

(3.3.1.3)

If (i) holds, then am/m3 = 0, hence µ(m2 ) = 1. Remark 3.1.2 gives

λ(aR) = µ(am) + µ(am2 ) + 1 = 2 = µ(m2 ) + 1

If (ii) holds, then µ(am) = µ(m2 ) − 1 by (3.3.1.3) and Remark 3.1.2 gives:

λ(aR) = µ(am) + µ(am2 ) + 1 = µ(m2 ) + 1

55 In either case, Remark 3.1.2 and (3.3.1.2) yield

λ(0 : a) = λ(R) − λ(aR) = µ(m2 ) + e + 2 − (µ(m2 ) + 1) = e + 1

Similar formulas hold for λ(bR) and λ(0 : b), hence

µ(m2 ) + 1 = λ(aR) = λ(0 : b) = e + 1

It follows that µ(m2 ) = e, thus HR (t) = 1 + et + et2 + t3 . We now establish the equivalence of conditions (3.1) and (3.2) for a, b in m r m2 . If (3.1) holds, then HR (−1) = 0 by the above argument. The equalities µ(am) = e − 1 = µ(bm) then follow from Proposition 3.1.3. Conversely, assume ab = 0 and µ(am) = e−1 = µ(bm). If am = 0 then µ(am) = e−1 = 0, contradicting the hypothesis e ≥ 3. If m3 = 0 then, as am 6= 0 and λ(0 :R m) = 1, one has am = m2 = (0 :R m) and µ(am) = 1. The equality µ(am) = e − 1 gives e = 2, contradicting the hypothesis. We have thus m3 6= 0. Since R is Gorenstein, one has (0 : m) = m3 and µ(m3 ) = 1. Note that am2 ⊆ m3 hence we have am2 = 0 or am2 = m3 . We will show that that am2 6= 0, implying that am2 = m3 . Similarly, bm2 = m3 and Proposition 3.1.3 shows that a, b is an exact pair of zero divisors. Indeed, assume am2 = 0. We then have am ⊆ (0 : m), hence am ⊆ m3 . If am = 0, then a ∈ (0 : m) = m3 , a contradiction. It remains thus that am = m3 , and hence µ(am) = 1. The equality µ(am) = e − 1 then yields e = 2, a contradiction. To establish the equivalence of (1) and (2) it is enough to recall that, since R is Gorenstein, one has (0 :R (0 :R I)) = I for every ideal I. It then remains to establish that (2) implies (3) (i), since the reverse implication is clear. For this, it is enough to show that a, b are in m r m2 . Note that 0 6= a ∈ m

56 implies 0 6= b ∈ m. As R is Gorenstein µ(m3 ) ≤ rankk (0 :R m) = 1, thus Lemma 3.1.4 (2) yields a, b ∈ m r m2 , concluding the equivalence of (2) and (3). Theorem 3.3.2. Let (R, m, k) be a local Gorenstein ring with µ(m) ≥ 3 and m4 = 0. If there exists a non-zero element a ∈ m such that the ideal (0 :R a) is principal, then the following hold: (1) Rg is Gorenstein; (2) R is Koszul; (3) every finite R-module M has a syzygy module that is Koszul. Proof. Proposition 3.3.1 shows that HR (t) = 1 + et + et2 + t3 with e = µ(m) and yields the existence of a element b such that a, b ∈ m r m2 form an exact pair of zero divisors. From Corollary 2.1, R = R/aR is Gorenstein with m3 = 0, thus (0 :R m) = m2 and g

µ(m2 ) = 1. It follows that (0 :Rg mg ) = mg2 and µ(mg2 ) = 1, hence R is Gorenstein as well. From Theorem 3.1.5(2) and (1), Rg /a∗ Rg is thus a Gorenstein local ring where a∗ is an exact zero divisor. It then follows from Corollary 2.1 that Rg is Gorenstein, establishing (1). Since rankk (0 :R m) = 1, the hypothesis µ(m) ≥ 3 establishes condition (2) of Theorem 3.2.3 concluding (2) and (3). Remark 3.3.3. The set of standard graded Gorenstein k-algebras R with HR (t) = 1 + et + et2 + t3 is in bijective correspondence with the set of degree 3 forms (up to scalars) in e variables over k, via the ‘inverse systems’ correspondence of Macaulay, as described in [24, §6]. As a consequence, such algebras can be parametrized by means of a projective space P(A3 ), where A3 is the degree three graded component of a polynomial ring over k in e variables. In view of Remark 3.2.5, the proof of [24, 6.4]

57 shows that when e ≥ 3 there exists a non-empty Zariski open subset of P(A3 ), whose points correspond to Gorenstein standard graded k-algebras admitting an exact pair of zero divisors (cf [24, 6.5]). In other words: a generic Gorenstein standard graded k-algebra of socle degree 3 has an exact pair of zero divisors in degree one. We now derive the numerical version of Theorem 3.3.2 as presented in the introduction. Corollary 3.3.4. Let (R, m, k) be Gorenstein with m4 = 0 and µ(m) = e ≥ 3. Assume there exist a non-zero element a ∈ m such that the ideal (0 :R a) is principal then the following hold: (1) HR (t) = 1 + et + et2 + t3 ; (2) HR (−t) PR k (t) = 1; (3) HR (−t) PR M (t) is in Z[t], for every finite R-module M . Proof. (1) It follows directly from Proposition 3.3.1 that HR (t) = 1 + et + et2 + t3 . (2) and (3) follow from the structural conditions (2) and (3) in Theorem 3.3.2 together with (2.4.3.1) and 3.2.2 respectively. To finish, we discuss the condition on µ(m) in the theorem. Any Gorenstein ring with µ(m) ≤ 2 is a complete intersection, see [42, Prop. 5], [13, §5], and Poincaré series over such rings are well understood. 3.3.5. Artinian complete intersections. Let (R, m, k) be an Artinian complete intersection ring, M be a finite module over R and let (R) denote the Hilbert-Samuel multiplicity of R. One always has:

−µ(m) (R) ≥ 2µ(m) , PR and (1 − t2 )µ(m) PR k (t) = (1 − t) M (t) ∈ Z[t],

(3.3.5.1)

58 (cf [17, §7, Prop. 7], [45, Thm. 6] and [29, 4.1]). Furthermore, the following conditions are equivalent: (1) R has minimal multiplicity, i.e. (R) = 2µ(m) , (2) R is Koszul, (3) HR (t) = (1 + t)µ(m) ; and when these conditions hold, one has

(1 − t)µ(m) PR M (t) ∈ Z[t].

(3.3.5.2)

Indeed, Avramov shows in [5, (2.3)] that (1) implies (2) and (3.3.5.2). The implication (2) ⇒ (3) follows from (2.4.3.1) and (3.3.5.1). On the other hand one has (R) = λ(R) = HR (1) as R is Artinian, so it becomes clear that (3) ⇒ (1).

59

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