ACYCLIC COMPLEXES OF FINITELY GENERATED FREE MODULES OVER LOCAL RINGS

ACYCLIC COMPLEXES OF FINITELY GENERATED FREE MODULES OVER LOCAL RINGS

MERI T. HUGHES, DAVID A. JORGENSEN, and LIANA M. ¸SEGA

Abstract

We consider the question of how minimal acyclic complexes of finitely generated free modules arise over a commutative local ring. A standard construction gives that every totally reflexive module yields such a complex. We show that for certain rings this construction is essentially the only method of obtaining such complexes. We also give examples of rings which admit minimal acyclic complexes of finitely generated free modules which cannot be obtained by means of this construction.

Introduction

Let R be a commutative local Noetherian ring with maximal ideal ᒊ. An acyclic complex of finitely generated freeR-modules is a complex

A: · · · →A₂−−→^d2 A₁−−→^d1 A₀−−→^d0 A₋₁−−→^d−1 A₋₂→ · · ·

with Ai finitely generated and free for eachi, and H(A) = 0. A complex Aas above is said to be totally acyclicor a complete resolution whenever H(A^∗) = 0, whereA^∗ = HomR(A, R). Complete resolutions are important in the study of maximal Cohen-Macaulay modules, and are used in defining Tate cohomology, cf. Avramov and Martsinkovsky [6] for a recent account and related problems. The properties and uses of complete resolutions have been studied extensively. More recently, the failure of acyclic complexes of free modules to be totally acyclic is studied by Jorgensen and ¸Sega in [14], and, in a more general setting, by Iyengar and Krause in [13]. Structure theorems for acyclic complexes of finitely generated free modules and the rings which admit non-trivial such complexes are given by Christensen and Veliche [11], in the caseᒊ³=0.

In this paper we consider the fundamental question of how acyclic com- plexes of finitely generated free modules arise. In order to ignore trivial in- stances, we restrict our attention tominimalcomplexesA, which are defined by the property thatdi(Ai)⊆ᒊAi−1for alli. A standard method of obtaining

Received November 29, 2007.

minimal acyclic complexes of finitely generated freeR-modules is through a process of dualization, described in Construction 2.2 below. The main object- ive of this paper is to demonstrate that although minimal acyclic complexes of finitely generated free modules often do arise by means of 2.2, this is not always the case, particularly when certain ring invariants are not too small.

For the purposes of this paper, we refer to complexes arising via Construc- tion 2.2 as sesqui-acycliccomplexes. These complexes are precisely those acyclic complexes of finitely generated free modulesAsatisfying Hi(A^∗)=0 for all i 0 (see Lemma 2.3). Totally acyclic complexes are thus sesqui- acyclic, and the examples in [14] show that the converse does not hold. As noted in [11], it was previously not known whether every minimal acyclic complex of finitely generated free modules is sesqui-acyclic. The main ex- amples of this paper show this is indeed not the case; they are given over local rings(R,_ᒊ)satisfying_ᒊ³=0, and thus complement the results of [11].

In Section 1 we identify classes of rings with the property that every minimal acyclic complex of finitely generated free modules is totally acyclic. It is well- known that Gorenstein rings are such. We prove, for example, an extension of this result for quotients of Golod rings by Gorenstein ideals.

Recall that thegeneralized Loewy lengthof a local ring(R,_ᒊ)is the integer (R) = min{n ≥ 0 | ᒊⁿ ⊆ (x) for some system of parametersxofR} and thecodimensionofR, denoted codimR, is the number edimR−dimR, where edimRdenotes the minimal number of generators ofᒊ. As a corollary of the result above, we show that every minimal acyclic complex of finitely generated free modules is totally acyclic wheneverRis Cohen-Macaulay, and one of the following conditions holds: (i) codimR≤2, (ii)(R)≤2, or (iii) codimR=(R)=3.

In Section 2 we give examples of minimal acyclic complexes of finitely generated free modules over Cohen-Macaulay ringsRwith codimR ≥5 and (R)≥3 which are not sesqui-acyclic. More precisely, we construct minimal acyclic complexesAof finitely generated freeR-modules with the property that Hi(A^∗)=0 for alli. The question whether every minimal acyclic complex of finitely generated free modules is sesqui-acyclic remains open for local rings with codimR =4 and(R)≥3.

1. Growth of Betti numbers and acyclic complexes

Throughout this section,Rdenotes a commutative local Noetherian ring with maximal idealᒊand residue fieldk.

1.1Complexes. We consider complexes of finitely generated freeR-mo- dules

A: · · · −→An+1 d_n+1^A

−−−→An d_n^A

−−→An−1−→ · · ·

Thetrivial complexis the complex withA_i =0 for alli.

The complexAis said to beacyclicif H(A) = 0. We let( )^∗ denote the functor HomR( , R). Thedual complex of Ais the complexA^∗, which has component(A_−n)^∗in degreen, and differentialsd_n^A∗ =(d₋^A_n+1)^∗. An acyclic complex of finitely generated free modulesAis said to be totally acyclicif H(A^∗)=0.

The complexAis said to beminimalifd_i^A(Ai)⊆ ᒊAi−1for alli∈^Z(cf.

[6, 8.1]).

For each integern, thenthsyzygy moduleofAisⁿA=Cokerd_n^A₊₁. In this section we identify several classes of rings which satisfy the property:

(a=ta)

Every minimal acyclic complex of finitely generated free R-modules is totally acyclic.

1.2 G-dimension. We recall the notion of G-dimension, introduced by Auslander and Bridger [1]. A finitely generated R-module M is said to be totally reflexive(or, equivalently, to haveG-dimension zero) if the following conditions hold:

(1) the natural evaluation mapM →M^∗∗is an isomorphism;

(2) Extⁱ_R(M, R)=0 for alli >0;

(3) Extⁱ_R(M^∗, R)=0 for alli >0.

A finitely generated R-module M is said to have Gorenstein dimension g, denoted G-dimRM = g, if g is the smallest integer such that there exists an exact sequence 0 → Gg → · · · → G1 → G0 → M → 0 with each Gi totally reflexive. If no such integer exists, then G-dimRM = ∞. The Auslander-Bridger formula for G-dimension states that if G-dimRM < ∞, then G-dimRM =depthR−depthM.

It is clear from the definitions that an R-module M is totally reflexive if and only ifM ∼= ⁰Afor some totally acyclic complex of finitely generated free modulesA. We collect below some variations on this fact, as suited for our purposes.

Lemma 1.3. Let A be an acyclic complex of finitely generated free R- modules. The following are then equivalent.

(1) Ais totally acyclic.

(2) ⁱAis totally reflexive for alli ∈^Z. (3) G-dimR(ⁱA) <∞for somei∈^Z.

Proof. (1)⇔(2): See Avramov and Martsinkovsky [6, Lem. 2.4].

Clearly, (2)⇒(3). To show (3)⇒(2), note that in a short exact sequence, if two modules have finite G-dimension then so does the third (see Auslander and Bridger [1, 3.11]). A recursive use of the short exact sequences:

0→^jA→A_j →^j−1A→0

gives then G-dimR(ⁱA) <∞for alli. The Auslander-Bridger formula and a counting of depths (by using the ‘depth lemma’ [7, 1.2.9]) along the short exact sequences above then give G-dimR(ⁱA)=0 for alli.

1.4Gorenstein rings. IfR is Gorenstein, then any finitely generated R- moduleM satisfies G-dimRM <∞ (see [1]). In consequence, Lemma 1.3 shows that every Gorenstein ringRsatisfies(a =ta). More generally, forR not necessarily local, Iyengar and Krause [13] show thatRis Gorenstein if and only if every acyclic complex of (not necessarily finitely generated) projective modules is totally acyclic.

In contrast to the result of [13], Gorenstein rings are not the only ones that are known to satisfy(a=ta). For example, ringsRwith_ᒊ²=0 are known to satisfy the property that every minimal acyclic complex of finitely generated freeR-modules is trivial. In Proposition 1.6 we generalize this result to a larger class of rings, characterized by a growth property for Betti numbers.

1.5Betti numbers. For every finitely generatedR-moduleMwe denote by β_i^R(M)theithBetti number ofM, defined to be the integer dimkTor^R_i (M, k);

it is equal to the rank of theith free module in a minimal free resolution ofM. IfAis a minimal acyclic complex of finitely generated freeR-modules, then we have:

(1.5.1) β_j^R_−i(ⁱA)=rankRA_j for allj andiwithj ≥i Moreover, the following statements are equivalent:

(1) Ais trivial

(2) A_i =0 for somei∈^Z.

(3) pd_RⁿA<∞for somen∈^Z.

Indeed, that (1)⇒(2)⇒(3) is clear. To justify (3)⇒(1), if pd_RⁿA<∞for somen, then pd_RⁱA<∞for alli < n. The Auslander-Buchsbaum formula for projective dimension and a count of depths gives then pd_RⁱA=0, hence ⁱAis free, for alli < n. Using minimality, it follows thatA_i = 0 for all i < n. Minimality again shows thatAi =0 for alli≥n.

Many local rings share the following growth property (see Remark 1.8 below):

()

⎧⎪

⎪⎨

⎪⎪

⎩

There exists a positive integerdsuch that for every finitely generatedR-moduleM of infinite projective dimension there exists a strictly increasing subsequence{β_n^R_i(M)}i≥0of

{β_i^R(M)}i≥0withid ≤n_i < (i+1)d for alli ≥0.

Proposition 1.6. If the local ring R satisfies () , then every minimal acyclic complex of finitely generated freeR-modules is trivial.

Proof. LetAbe a minimal acyclic complex, and set m=max{rankRA_i |0≤i < d},

wheredis as in the definition of(). DefineM =^−mdA. If pd_RM = ∞, then there exists a subsequence{β_ni(M)}of{β_n(M)}such thatβ_ni(M)≥β_n0(M)+ i, andid≤n_i < (i+1)dfor alli ≥0. In particular,β_nm(M)≥β_n0(M)+m withmd ≤n_m< (m+1)d. In other words,

rankRA_nm−md ≥rankRA_n0−md +m withmd ≤nm< (m+1)d.

Since rankRA_n0−md > 0, we see that rankRA_nm−md > m. But this con- tradicts the definition ofm, as 0 ≤n_m−md < d. Therefore it must be that pd_RM <∞, henceAis trivial (see 1.5).

Recall that anR-moduleM is said to beperfectif pd_RM =grade_RM. An idealI ofRis said to beperfect if theR-moduleR/I is perfect. The idealI is said to beGorensteinif it is perfect andβ_g^R(R/I )=1 forg =grade_RI.

Since regular local rings satisfy property()vacuously, the following the- orem is a generalization of the fact mentioned in 1.4, that Gorenstein rings satisfy(a=ta).

Theorem1.7. AssumeRsatisfies property(). LetIbe a Gorenstein ideal ofRand setS=R/I. The local ringSsatisfies then(a=ta).

The theorem will be proved at the end of the section. We proceed now to give some applications.

Remark1.8. It is known thatRsatisfies()in all of the following cases:

(a) Ris a regular local ring.

(b) Ris a Golod ring which is not a hypersurface. (Peeva, [17, Proposition 5]).

(c) R =S/_ᒋJ, where(S,_ᒋ)is a local ring andJ is an ideal ofSsuch that rankkJ /_ᒋJ ≥2 (Lescot, [15, Theorem A.2]).

(d) R=S/_ᒋJ, where(S,_ᒋ)is a local ring andJ is anᒋ-primary ideal ofS (Lescot, [15, Theorem A.1]).

(e) R = S/I, where(S,_ᒋ)is a local ring andI is an ideal ofSsuch that rankk(_ᒋJ /_ᒋI )≥2, whereJ =(I:Sᒋ)(Choi, [9, Theorem 1.1]).

(f) ᒊ³ = 0, and eithere := rankkᒊ/_ᒊ² < rankkᒊ², orP_k^R(t ) = (1− et+(e−1)t²)⁻¹(Lescot, [15, Theorem B(1)]).

In view of Remark 1.8, Theorem 1.7 identifies many classes of rings which satisfy(a=ta). We can easily extend these classes even further by means of usual homological constructions.

Proposition1.9. IfAis a complex of finitely generated freeR-modules, R → Sis a homomorphism of local rings, andxis anR-regular sequence, then the following hold:

(1) Ais acyclic (respectively, totally acyclic) if and only ifA⊗RR/(x)is acyclic (respectively, totally acyclic).

(2) If R → S is a faithfully flat, thenA is acyclic (respectively, totally acyclic) if and only ifA⊗RSis acyclic (respectively, totally acyclic).

Proof. It suffices to show thatAis acyclic if and only ifA⊗RR/(x)and A⊗RSare acyclic. The statements for total acyclicity follow in view of the isomorphisms of complexes

HomR/(x)(A⊗RR/(x), R/(x))∼=A^∗⊗RR/(x), and HomS(A⊗RS, S)∼=A^∗⊗RS.

Statement (2) is obvious. To prove (1), we may assume thatxconsists of a single regular elementx. Note that there exists an exact sequence of complexes 0→A−→^x A→A⊗^RR/(x)→0 which gives rise in homology to the exact sequence

· · · →Hi(A)−→^x Hi(A)→Hi(A⊗RR/(x))→Hi−1(A)−→ · · ·^x Obviously Hi(A)=0= Hi−1(A)implies Hi(A⊗RR/(x))= 0. Also, since Hi(A)is finitely generated for eachi, Nakayama’s lemma gives that if Hi(A⊗R

R/(x))=0, then Hi(A)=0.

The following is an easy corollary of Proposition 1.9.

Corollary1.10. The following hold for a local ringR:

(1) Ifxis a regularR-sequence andR/xRsatisfies(a=ta), thenRsatisfies (a=ta).

(2) If there exists a flat homomorphismR →Sof local rings, andSsatisfies (a=ta), thenRsatisfies(a=ta).

We have the following corollary of Proposition 1.6/Theorem 1.7, which indicates to what extent the examples of Section 3 are minimal with respect to the invariants Loewy length and codimension.

Corollary1.11. LetRbe a local Cohen-Macaulay ring of codimension cand generalized Loewy length. ThenRsatisfies(a= ta), provided one of the following conditions holds:

(1) c≤2 (2) ≤2

(3) c=mult(R)−1.

(4) c==3

Proof. In view of Remark 1.4, we may assume that the ring R is not Gorenstein.

Under the assumption in (1), Scheja [18] proves that the ringRis either a complete intersection, or a Golod ring. Since we assumedRis not Gorenstein, R is thus Golod. The result follows from Proposition 1.6 and Peeva’s result mentioned in Remark 1.8(b).

(2) In this case there exists a regular sequence x such that the square of the maximal ideal ofR/xRis zero. A local ring for which the square of the maximal ideal is zero is Golod. Since the elements of the regular sequence can assumed to be inᒊ−ᒊ² it follows from [3, 5.2.4(2)] that R is also Golod.

The result thus follows from Proposition 1.6 and 1.8(b).

(3) In this caseRhas minimal multiplicity, and is therefore Golod (see [3, 5.2.8]). Thus again the result follows from Proposition 1.6 and 1.8(b).

(4) By hypothesis there exists a maximalR-sequenceysuch that the ring S = R/yR satisfiesᒋ³ = 0, whereᒋis the maximal ideal ofS. Theorem A of [11] gives that the Poincaré seriesP_k^S(t )=

β_i^S(k)tⁱ ofkoverSsatisfies P_k^S(t )=(1−3t+2t²)⁻¹, and thus has a pole of order 1 att =1. A result of Avramov [2, Theorem 3.1] shows that the ringShas an embedded deformation, that is, there exists a local ring(R,_ᒊ)and a regular elementx∈(_ᒊ)²such thatS ∼= R/xR. Furthermore, we have codimR = 2. Since we assumed thatR is not Gorenstein, neither areSorR, henceRis a Golod ring, as in the proof of part (1). Note that the idealxRis Gorenstein, sincexis a regular element. The ringSsatisfies then(a=ta)by Theorem 1.7 and Remark 1.8(b).

Corollary 1.10(1) gives then the conclusion forR.

We now prepare for the proof of Theorem 1.7. For the convenience of the reader, we provide a proof of the following well-known inequality:

Lemma1.12. LetI ⊆ Rbe an ideal. SetS= R/I and letM be a finitely generatedS-module. The following inequality holds for alln≥0:

β_n^R(M)≤

p+q=n

β_q^R(S)β_p^S(M)

Proof. Consider the change of rings spectral sequence (see [8, XVI §5]) E_p,q² =Tor_p^S M,Tor^R_q(S, k)

⇒Tor_p^R_+q(M, k)

Note that E_p,q^r+1is a subquotient of E_p,q^r for allr ≥2 and the spaces E_p,q^∞ are the subfactors of a filtration of Tor_p^R_+q(M, k). We have thus (in)equalities:

rankkTor^R_n(M, k)=

p+q=n

rankkEp,q^∞ ≤

p+q=n

rankkEp,q² =

p+q=n

β_q^R(S)β_p^S(M)

Proof of Theorem 1.7. Set g = gradeI, and let A be a non-trivial minimal acyclic complex of finitely generated freeS-modules. Set

m= max

0j <d

^g

i=0

β_i^R(S)rankSAj−i

,

where the integerd is defined by (), andM = ^−mdA. IfM has infinite projective dimension overR, then property()gives a subsequence{β_n^R

i(M)} of{β_i^R(M)}such that β_n^R_m(M) ≥ β_n^R₀(M)+mand md ≤ nm ≤ (m+1)d.

Now setj =nm−md. Then 0≤j < d, and by the definition ofmwe have β_n^R

m(M)≥β_n^R

0(M)+

g i=0

β_i^R(S)rankSA_j−i.

Sinceβ_n^R₀(M) >0 and rankSA_t = β_t^S_+md(M)for allt ≥ −md, in particular rankSA_j−i =β_n^S

m−i(M)for 0≤i≤g(sinceg≤m≤n_m), we obtain β_n^R_m(M) >

g i=0

β_i^R(S)β_n^S_m−i(M)

and this contradicts the formula given by Lemma 1.12. In consequence, we have that pd_RM <∞. We have thus G-dimRM <∞and hence G-dimSM <

∞, by [12, Prop. 5] (cf. also [4, 7.11]). Using Lemma 1.3, this implies thatA is totally acyclic.

2. Sesqui-acyclic Complexes of Free Modules and Examples

Unless otherwise specified, R denotes a commutative local Noetherian ring with maximal idealᒊand residue fieldk. We use the terminology and notation of Section 1 for complexes.

Definition. We say that a complexAissesqui-acyclicif it is acyclic and there exists an integercsuch that Hi(A^∗)=0 for alli > c.

2.1Comparison of acyclic, sesqui-acyclic and acyclic complexes. Clearly, every totally acyclic complex is sesqui-acyclic, but the converse does not hold.

Indeed, Jorgensen and ¸Sega in [14] construct minimal acyclic complexes of finitely generated free R-modulesC with the property that Hi(C^∗) = 0 if and only if i ≥ 1. We thus have the following diagram of implications for complexes of finitely generated freeR-modules (with the right implication also following directly from the definitions):

(2.1.1) totally acyclic sesqui-acyclic acyclic

In this section we first give motivation for the definition of sesqui-acyclic complex, and then we show that the right-hand implication in (2.1.1) is not re- versible. More precisely, we construct minimal acyclic complexesAof finitely generated free modules over codimension five local ringsRwithᒊ³=0 such that Hi(A^∗)=0 for alli. We then extend these to such examples over Cohen- Macaulay ringsRwhere any choice of codimR ≥5 and(R)≥codimR−2 is allowed. Note that Corollary 1.11 shows that certain restrictions on the codi- mension and generalized Loewy length are necessary, hence our examples are minimal, at least with respect to generalized Loewy length. We do not know whether such examples can be constructed when codimR =4.

As seen in Proposition 1.6, non-trivial minimal acyclic complexes of finitely generated free R-modules may not exist. When they do exist, we want to understand how they arise. A standard construction is described below (cf.

[11, 3.2]).

Construction2.2. Suppose thatMis anR-module satisfying (2.2.1) Extⁱ_R(M, R)=0 for all i >0.

LetP−→^π M^∗be a free resolution ofM^∗, with P:· · · −→P2

d₂^P

−−→P1 d₁^P

−−→P0→0,

andQ−→^η Mbe a free resolution ofM, with Q:· · · −→Q₂ ^d

Q

−−→2 Q₁ ^d

Q

−−→1 Q₀→0.

By condition (2.2.1), the complex 0−→M^∗−−→^η∗ Q^∗₀ ^(d

Q 1)^∗

−−−→Q^∗₁ ^(d

Q 2)^∗

−−−→Q^∗₂−→ · · ·

is exact, and thus one can splice the complexes Pand Q^∗ = HomR(Q, R) together to obtain an acyclic complex of free modules:

P|Q^∗:· · · −→P₂ ^d

P

−−→2 P₁ ^d

P

−−→1 P₀−−−→^η∗◦π Q^∗₀ ^(d

Q 1)^∗

−−−→Q^∗₁ ^(d

Q 2)^∗

−−−→Q^∗₂−→ · · · with the convention that(P|Q^∗)_i = P_i fori ≥0, and(P|Q^∗)_i = Q^∗_−i−1for i < 0. This complex is minimal wheneverPandQare chosen minimal and M has no non-zero free direct summand. It is non-trivial ifM is non-zero.

Recall that thenthshiftof the complexAis the complex ⁿAwith( ⁿA)_i = A_i−nandd_i ^nA=(−1)ⁿd_i^A_−n. We writeA_nfor the complex withith compon- ent (respectively, differential) equal toA_i (respectively,d_i^A) ifi≥n(respect- ively,i > n) and 0 ifi < n(respectively,i ≤n).

Lemma 2.3. Let A be an acyclic complex of finitely generated free R- modules. The following are equivalent:

(1) Ais sesqui-acyclic.

(2) There exist an integersand anR-moduleMsatisfying(2.2.1)such that Ais isomorphic to ^s(P|Q^∗), where the complexP|Q^∗is defined as in Construction2.2, withPa free resolution ofM^∗andQa free resolution ofM.

Proof. (1)⇒(2). Assume thatAsatisfies Hi(A^∗)=0 for alli > c. Letsbe any integer satisfyings≥cand setM =^s(A^∗). ThenQ= ^−s((A^∗)_s)is a free resolution ofM. SinceAis exact, we have Hi(Q^∗)=0 for alli <0, hence Extⁱ_R(M, R)=0 for alli >0. Note that we haveM^∗ =^−s+1A, and hence P= ^s−1(A_−s+1)is a free resolution ofM^∗, and thusA∼= ^−s+1(P|Q^∗).

(2)⇒(1). Suppose thatA ∼= ^s(P|Q^∗) for somes, withP|Q^∗ as in 2.2.

One has

Hi(A^∗)∼=Hi+s−1(Q^∗∗)∼=Hi+s−1(Q)=0 for all i >−s.

Lemma 2.3 thus shows that sesqui-acyclic complexes are precisely those acyclic complexes which come about through a process of dualization. In [11,

3.2], Christensen and Veliche noted that all known examples of minimal acyclic complexes of finitely generated freeR-modules arise by means of this process of 2.2. The authors also raised the question in [11, 3.4] of whether this is always the case, or in terms of our discussion, whether the right-hand implication in (2.1.1) is reversible. We now show that it is not reversible.

From now on, the ringRand the complexAare as defined below.

2.4The main example. Letk be a field andα ∈kbe non-zero. Consider the quotient ring

R=k[X1, X2, X3, X4, X5]/I

where theX_i are indeterminates (each of degree one), andI is the ideal gen- erated by the following 11 homogeneous quadratic relations:

X²₁, X²₄, X₂X₃, αX₁X₂ +X₂X₄, X₁X₃+X₃X₄, X₂², X2X5−X1X3, X₃²−X1X5, X4X5, X²₅, X3X5

As a vector space overk,Rhas a basis consisting of the following 10 elements:

1, x1, x₂, x₃, x₄, x₅, x₁x₂, x₁x₃, x₁x₄, x₁x₅

wherexi denote the residue classes ofXi moduloI. SinceI is generated by homogeneous elements, R is graded, and has Hilbert series 1+5t + 4t². MoreoverR has codimension five, and it is local with maximal idealᒊ = (x₁, . . . , x₅)satisfyingᒊ³=0.

For each integer i ∈ ^Z we let di:R² → R² denote the map given with respect to the standard basis ofR²by the matrix

x1 αⁱx2

x3 x4

Consider the sequence of homomorphisms:

A: · · · →R²−−−→^di+1 R²−→^di R²−−−→^di−1 R²→ · · ·

Theorem2.5. The sequenceAis a minimal acyclic complex of free R- modules withHi(A^∗)=0for alli∈^Z.

We have thus:

Corollary2.6. The minimal acyclic complexAis not sesqui-acyclic.

Remark2.7. Whenα ∈ k is an element of infinite multiplicative order, the complexAis non-periodic. Whenα has multiplicative orders for some integers >0, one has thatAis periodic of periods.

Remark2.8. Christensen and Veliche ask in [11, 3.5] whether every acyclic complex of free modules Cwith {rankC_i} constant, over a local ring with ᒊ³=0, is totally acyclic. Theorem 2.5 gives a negative answer to this question.

Proof of Theorem2.5. Using the defining relations ofR, one can easily show thatd_id_i+1=0 for alli, henceAis a complex.

We let(a, b)denote an element ofR²written in the standard basis ofR² as a freeR-module. For eachi, thek-vector space Imd_i is generated by the elements:

d_i(1,0)=(x₁, x₃) d_i(0,1)=(αⁱx₂, x₄) d_i(x₁,0)=(0, x1x₃) d_i(x₂,0)=(x₁x₂,0) d_i(x₃,0)=(x₁x₃, x₁x₅) d_i(x₄,0)=(x₁x₄,−x₁x₃)

d_i(x₅,0)=(x₁x₅,0) d_i(0, x1)=(αⁱx₁x₂, x₁x₄) d_i(0, x2)=(0,−αx₁x₂) d_i(0, x3)=(0,−x₁x₃) d_i(0, x4)=(−αⁱ⁺¹x₁x₂,0) d_i(0, x5)=(αⁱx₁x₃,0)

Excludingdi(0, x3)anddi(0, x4), the above equations provide 10 linearly independent elements in Imdi. Thus rankk(Imdi)=10 for alli. Since

rankkKerd_i +rankkImd_i =rankkR²=20

we have dim Kerdi =10 for alli. Thus, Imdi+1 =Kerdi for alli, so thatA is acyclic.

To prove Hi(A^∗) = 0, we have thatd_i^∗ = (d_i)^∗:R² →R²is represented with respect to the standard basis ofR²by the matrix

x₁ x₃ αⁱx2 x4

For eachi, the vector space Imd_i^∗is generated by the following elements d_i^∗(1,0)=(x1, αⁱx2)

d_i^∗(0,1)=(x3, x4) d_i^∗(x1,0)=(0, αⁱx1x2) d_i^∗(x2,0)=(x1x2,0) d_i^∗(x3,0)=(x1x3,0)

d_i^∗(x4,0)=(x1x4,−αⁱ⁺¹x1x2)

d_i^∗(x5,0)=(x1x5, αⁱx1x3) d_i^∗(0, x1)=(x1x3, x1x4) d_i^∗(0, x2)=(0,−αx1x2) d_i^∗(0, x3)=(x1x5,−x1x3) d_i^∗(0, x4)=(−x1x3,0) d_i^∗(0, x5)=(0,0)

Excludingd_i^∗(0, x2),d_i^∗(0, x4), andd_i^∗(0, x5)which are redundant, we have only 9 linearly independent elements in Imd_i^∗, hence rankkImd_i^∗=9 for every i. It follows that rankkKerd_i^∗=11 for alli, hence Hi(A^∗)=0.

One can easily get examples of minimal acyclic complexes which are not sesqui-acyclic over local rings of any codimension larger than five as follows.

Letn≥1 be an integer, andy1, . . . , ynbe indeterminates overk. DefineRn

to be the local ring obtained by localizingR⊗kk[y1, . . . , yn] at the maximal ideal

ᒊn=(x_i⊗1,1⊗y_j |1≤i ≤5,1≤j ≤n) Now letA_ndenote the sequence

· · · →R²_n ^d

i+1n

−−−→R_n^{2 d}

n

−−→i R_n^{2 d}

i−1n

−−−→R_n²→ · · · whered_iⁿdenotes the mapd_i⊗kk[y1, . . . , y_n] localized atᒊn.

Letp₁, . . . , p_n be positive integers≥ 2, set = n

i=1(p_i −1)+3, and consider theR_n-sequencey=1⊗y₁^p1, . . . ,1⊗y_n^pn.

Corollary 2.9. The ring R_n is a local Cohen-Macaulay ring with codim(Rn) = 5, dim(Rn) = n and (R_n) = 3, and S_n = R_n/(y) is an artinian local ring withcodim(Sn) = n+5and(S_n) = , such that the following hold:

(1) A_nis a minimal acyclic complex of freeRn-modules which is not sesqui- acyclic.

(2) A_n⊗Rn Rn/(y)is a minimal acyclic complex of freeSn-modules which is not sesqui-acyclic.

Proof. The statements aboutRnandSnare clear.

SinceR→R⊗kk[y1, . . . , yn] is a faithfully flat embedding of rings, the complexA⊗kk[y1. . . , y_n] stays exact, and then too after localizing. ThusA_n is acyclic (and obviously minimal).

We have Hi(A^∗)⊗k k[y1, . . . , y_n] ∼= Hi(A^∗⊗k k[y1, . . . , y_n]) for alli.

Moreover,A^∗⊗kk[y1, . . . , yn] localized atᒊnis isomorphic to(A_n)^∗. Since Hi(A^∗) = 0 for alli by Theorem 2.5, it follows that Hi((A_n)^∗) = 0 for all i ∈ ^Z. HenceA_n is not sesqui-acyclic. The statement aboutA_n⊗Rn R_n/(y) follows from Proposition 1.9.

In summary, we have the following diagram of implications for complexes of finitely generated free modules over a local ringR:

totally acyclic sesqui-acyclic acyclic

Acknowledgements. We would like to thank the referee for his/her valu- able suggestions on an earlier version of this paper, particularly for those re- garding Section 1. We would also like to thank Lars Christensen, and Hamid Rahmati for their helpful comments.

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MERI T. HUGHES

DEPARTMENT OF MATHEMATICS UNIVERSITY OF TEXAS AT ARLINGTON ARLINGTON, TX 76019

U. S. A.

E-mail:meri@uta.edu

DAVID A. JORGENSEN DEPARTMENT OF MATHEMATICS UNIVERSITY OF TEXAS AT ARLINGTON ARLINGTON, TX 76019

U. S. A.

E-mail:djorgens@uta.edu

LIANA M. ¸SEGA

DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MISSOURI-KANSAS CITY KANSAS CITY, MO 64110

U. S. A.

E-mail:segal@umkc.edu