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ON THE VANISHING OF HOMOLOGY WITH MODULES OF FINITE LENGTH

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ON THE VANISHING OF HOMOLOGY WITH MODULES OF FINITE LENGTH

PETTER ANDREAS BERGH

Abstract

We study the vanishing of homology and cohomology of a module of finite complete intersection dimension over a local ring. Given such a module of complexityc, we show that ifc(co)homology groups with a module of finite length vanish, then all higher (co)homology groups vanish.

1. Introduction

With his 1961 paper [1], Auslander initiated the study of vanishing of homology for modules over commutative Noetherian local rings. In that paper, he proved his famous rigidity theorem: ifM andN are finitely generated modules over an unramified regular local ringA, then the implication

TorAn(M, N)=0 ⇒ TorAi (M, N)=0 for in

holds for anyn≥0. Lichtenbaum settled the ramified case in [10], and Murthy generalized this to arbitrary complete intersection rings in [11]. Namely, he proved that ifM andN are finitely generated modules over a local complete intersectionAof codimensionc, and

TorAn(M, N)=TorAn+1(M, N)= · · · =TorAn+c(M, N)=0

for somen≥0, then TorAi (M, N)=0 forin. This was vastly generalized in [8], [9], where Jorgensen focused on thecomplexitiesof the modules involved, rather than the codimension of the ring. He showed that ifM is a module of finite complete intersection dimension and complexitycover a local ring, then the vanishing ofc+1 homology (respectively, cohomology) groups forces the vanishing of all the higher homology (respectively, cohomology) groups.

In all of the above mentioned vanishing results, one assumes the vanishing ofconsecutive(co)homology groups. However, the author showed in [4] that this is not necessary. In fact, the (co)homology groups assumed to vanish may be arbitrarily far apart from each other. Namely, letMandNbe modules over a

Received 1 December 2009.

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local ringA, withMof finite complete intersection dimension and complexity c. It was shown that if there exists an odd numberqsuch that

TorAn(M, N)=TorAn+q(M, N)= · · · =TorAn+cq(M, N)=0

for somen >depthA−depthM, then TorAi (M, N)= 0 fori >depthA− depthM (and similarly for cohomology). The above vanishing result of Jor- gensen is the special caseq =1.

In this paper, we show that when the moduleN has finite length, then we may reduce the number of (co)homology groups assumed to vanish by one.

Namely, letM andN be modules over a local ringA, withN of finite length and M of finite complete intersection dimension and complexityc. In this situation, we show that if there exists an odd numberqsuch that

TorAn(M, N)=TorAn+q(M, N)= · · · =TorAn+(c−1)q(M, N)=0 for somen >depthA−depthM, then TorAi (M, N)= 0 fori >depthA− depthM(and similarly for cohomology). The special case whenq=1 and the ring is a complete intersection was proved by Jorgensen in the above mentioned papers.

2. Complete intersection dimension

Throughout this paper, we assume that all modules encountered are finitely generated. In this section, we fix a local (meaning commutative Noetherian local) ring(A,, k).

EveryA-moduleM admits a minimal free resolution

· · · →F2F1F0M →0

which is unique up to isomorphism. The rank of the freeA-moduleFnis the nthBetti numberofM; we denote it byβnA(M). ThecomplexityofM, denoted cxM, is defined as

cxM def= inf{t ∈N∪ {0} | ∃a∈Rsuch thatβnA(M)ant−1for alln0}.

Thus, the complexity of a module is the polynomial rate of growth of its Betti sequence.

An arbitrary local ring may have many modules with infinite complexity. In fact, by a theorem of Gulliksen (cf. [7, Theorem 2.3]), a local ring is a complete intersection if and only if all its modules have finite complexity. In order to study modules behaving homologically as modules over such rings, Avramov, Gasharov and Peeva defined in [2] the notion of modules with finitecomplete

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intersection dimension. Recall that a quasi-deformation of A is a diagram ARQof local homomorphisms, in whichAR is faithfully flat, andRQis surjective with kernel generated by a regular sequence. The length of this regular sequence is thecodimensionof the quasi-deformation.

An A-module M has finite complete intersection dimension if there exists such a quasi-deformation in which theQ-moduleRAMhas finite projective dimension. We shall write “CI-dimension” instead of “complete intersection dimension”.

As mentioned, the homological behavior of modules of finite CI-dimension reflects the behavior of modules over local complete intersections. For ex- ample, such a module has finite complexity. Moreover, the cohomology groups are finitely generated over a ring of cohomology operators of degree two, a notion we now explain (cf. [2, 4.1]). LetQbe a local ring, and letx1, . . . , xc

be a regular sequence of lengthc. Denote the ringQ/(x1, . . . , xc)byR, and letM be anR-module with a free resolutionF. The regular sequence gives rise to chain maps{ti ∈HomR(F,F)}ci=1of degree−2, namely theEisenbud operators. These are uniquely defined up to homotopy, and are therefore ele- ments of Ext2R(M, M). Moreover, these chain maps commute up to homotopy.

Thus, the Eisenbud operators give rise to a polynomial ringR[χ1, . . . , χc] of operators with the following properties:

(1) The degree ofχi is two for alli.

(2) There is a graded ring homomorphismR[χ1, . . . , χc]−→ϕM ExtR(M, M) for everyR-moduleM, where ExtR(M, M)= ⊕n=0ExtnR(M, M). (3) For every pair(X, Y )ofR-modules, theR[χ1, . . . , χc]-module structure

on ExtR(X, Y )throughϕXandϕY coincide (cf. [3, Theorem 3.3]).

(4) For every pair(X, Y )ofR-modules, theR[χ1, . . . , χc]-module ExtR(X, Y )is finitely generated whenever ExtnQ(X, Y ) = 0 forn 0 (cf. [6, Theorem 3.1]).

As mentioned above, modules of finite CI-dimension have finitely generated cohomology, as we see from property (4) above. Namely, letM andN beA- modules, withMof finite CI-dimension over a quasi-deformationARQ, say. Furthermore, let S be the polynomial ring induced by the Eisenbud operators coming from the surjectionQR. Then since pdQ(RAM)is finite, theS-module ExtR(RAM, RAN)is finitely generated.

Having defined Eisenbud operators, we end this section with the following result. It shows that a module of finite CI-dimension and complexity one can be “realized” by a codimension one quasi-deformation.

Lemma 2.1. Let A be a local ring, and letM be an A-module of finite

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CI-dimension and complexity one. Then there exists a codimension one quasi- deformationARQwithpdQ(RAM) <.

Proof. By [2, Proposition 7.2(2)], there exists a quasi-deformationARQof codimension one, such that the Eisenbud operatorχon the minimal free resolution ofRAMis eventually surjective. Thus, if

· · · →F2F1F0M →0 is the minimal free resolution ofM, and

· · · −−→RAFn+2−−→RAFn+1−−→ RAFn −−→ · · ·

fn+2 fn+1 fn

· · · −−→ RAFn −−→RAFn−1−−→RAFn−2−−→ · · · is the chain map corresponding toχ, thenfn is surjective for largen. By [2, Theorem 7.3], the minimal free resolution ofMis eventually periodic of period 2, so thatFnis isomorphic toFn+2forn0. Consequently, the mapfnis an isomorphism for largen.

For anyR-moduleN and integern, the action ofχ on ExtnR(RAM, N) is induced byfn. Therefore, there exists an integern0such that

χExtnR(RAM, N)=Extn+R 2(RAM, N)

for allnn0. Consequently, theR[χ]-module ExtnR(RAM, N)is Noeth- erian, and so it follows from [2, Theorem 4.2] that ExtnQ(RAM, N)vanishes for largen. In particular, this holds if we takeN to be the residue field ofQ, hence pdQ(RAM)is finite.

3. Vanishing of homology

We start by showing the following special case of the main result. It shows that when a module has finite CI-dimension and complexity one, then the vanishing of a single homology group with a module of finite length implies the vanishing of all higher homology groups.

Proposition 3.1. Let A be a local ring, and let M be an A-module of finite CI-dimension and complexity one. Furthermore, letN be anA-module of finite length. IfTorAn(M, N) = 0for some n > depthA−depthM, then TorAn(M, N)=0for alln >depthA−depthM.

Proof. Choose, by Lemma 2.1, a codimension one quasi-deformation ARQfor which the projective dimension of theQ-moduleRAMis finite. ThusR =Q/(x)for a regular elementxinQ. Denote theR-modules

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RAMandRANbyMandN, respectively, and note thatNhas finite length.

The change of rings spectral sequence TorpR

M,TorQq(N, R)

p Torp+qQ (M, N) degenerates into a long exact sequence

... ... ...

TorR1(M, N)−−→TorQ2(M, N)−−→TorR2(M, N)−−→

TorR0(M, N)−−→TorQ1(M, N)−−→TorR1(M, N)−−→

in homology. Let

0→QβdQ(M)→ · · · →Qβ0Q(M)M→0

be a minimalQ-free resolution ofM. Localizing this resolution atx, keeping in mind thatxM=0, we see thatd

i=0(−1)iβiQ(M)=0. Now(TorQi (M, l)) = βiQ(M), where l is the residue field ofQ. Moreover, the Euler char- acteristic

i=0(−1)i(TorQi (M,−))is well defined and additive on the cat- egory of finite lengthQ-modules. Therefore, induction on length shows that

i=0(−1)i(TorQi (M, X))=0 for everyQ-module of finite length.

By the Auslander-Buchsbaum formula, the projective dimension ofMover Qis given by

depthQQ−depthQM=depthRR−depthRM+1

=depthAA−depthAM +1.

Let m > depthAA− depthAM be an integer, and consider the long exact sequence in homology starting in TorQm+1(M, N) = 0, i.e. the long exact sequence obtained from the above spectral sequence. By taking the altern- ating sum of the lengths of the terms, and identifying TorR0(M, N) with TorQ0(M, N), we see that(TorRm(M, N))=(TorRm+1(M, N)).

Suppose that TorAn(M, N) = 0 for somen > depthA−depthM. Since TorRn(M, N) is isomorphic to RA TorAn(M, N), the above shows that TorRn(M, N)= 0 for alln > depthA−depthM. Then by faithful flatness, the homology group TorAn(M, N)vanishes for alln >depthA−depthM.

Having proved the complexity one case, we now prove the main result, the case when one of the modules has finite CI-dimension and complexityc, and the other has finite length. In this situation, the vanishing ofchomology

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groups implies the vanishing of all higher homology groups. Moreover, the

“vanishing gap” can be any odd number. This result therefore generalizes [4, Theorem 3.5] to rings which are not necessarily complete intersections.

Theorem3.2. LetAbe a local ring, and letM be anA-module of finite CI-dimension and complexityc. Furthermore, letNbe anA-module of finite length. If there exist an odd numberq ≥ 1and a number n > depthA− depthMsuch thatTorAi (M, N)=0fori ∈ {n, n+q, . . . , n+(c−1)q}, then TorAi (M, N)=0for alli >depthA−depthM.

Proof. We argue by induction on the complexity cofM. Ifc = 0, then pdM = depthA−depthM by the Auslander-Buchsbaum formula, and the result follows. The casec=1 is Proposition 3.1, so we assume thatc≥2.

By [5, Lemma 3.1], there exists a faithfully flat extensionARof local rings with the following properties:

(1) There is an exact sequence

0→RAMKqR(RAM)→0 ofR-modules.

(2) TheR-modulesRAM andKhave finite CI-dimension.

(3) The complexity ofKisc−1.

(4) depthRR−depthRK=depthAA−depthAM.

As in the previous proof, denote theR-modulesRAM andRANbyM andN, respectively, and note thatNhas finite length. Since TorRi (M, N)is isomorphic toRATorAi (M, N)for alli, the vanishing assumption implies that TorRi (M, N) = 0 fori ∈ {n, n+q, . . . , n+(c−1)q}. From (1), we obtain a long exact sequence

· · · →TorRi+q+1(M, N)→TorRi (M, N)

→TorRi (K, N)→TorRi+q(M, N)→ · · · of homology groups, from which we obtain that TorRi (K, N) = 0 for i ∈ {n, n+q, . . . , n+(c−2)q}. Therefore by induction and properties (2), (3) and (4), the homology group TorRi (K, N)vanishes for alli > depthAA− depthAM. The long exact sequence then shows that TorRi (M, N) TorRi+j (q+1)(M, N)for alli >depthAA−depthAM andj ≥0.

By considering the pairs (i, j ) ∈ {(n, c), (n+q, c−1), . . . , (n+(c− 1)q,1)}, we see that TorRi (M, N)=0 forn+cq+1≤in+cq+c. Faithful flatness then implies that TorAi (M, N)=0 forn+cq+1≤in+cq+c. Thus we have reduced to the case whencconsecutivehomology groups vanish.

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Now we use [5, Lemma 3.1] once more: there exists a faithfully flat local homomorphismASand an exact sequence

0→SAMK1S(SAM)→0

of Smodules, in which SAM and K satisfy properties (2), (3) and (4).

Arguing as above, we obtain an isomorphism

TorSi(SAM, SAN)TorSi+2(SAM, SAN)

for all i > depthAA− depthAM. Since TorSi(SA M, SA N) = 0 for n+ cq + 1 ≤ in+cq + c and c is at least two, we conclude that TorSi(SAM, SAN)=0 for alli >depthAA−depthAM. Finally, faithful flatness implies that TorAi (M, N)=0 for alli >depthAA−depthAM.

We record the special caseq = 1 in the following corollary, i.e. the case whencconsecutive homology groups vanish. Note that the case when the ring is a complete intersection follows from [8, Theorem 2.6].

Corollary 3.3. Let A be a local ring, and let M be an A-module of finite CI-dimension and complexityc. Furthermore, let N be an A-module of finite length. If there exists a number n > depthA−depthM such that TorAi (M, N) = 0 for nin+ c−1, then TorAi (M, N) = 0 for all i >depthA−depthM.

We also include the cohomology versions of Theorem 3.2 and Corollary 3.3.

We do not include a proof; the proofs of Proposition 3.1 and Theorem 3.2 carry over verbatim to the cohomology case.

Theorem3.4.LetAbe a local ring, and letM be anA-module of finite CI-dimension and complexityc. Furthermore, letNbe anA-module of finite length. If there exist an odd numberq ≥ 1and a number n > depthA− depthMsuch thatExtiA(M, N)=0fori∈ {n, n+q, . . . , n+(c−1)q}, then ExtiA(M, N)=0for alli >depthA−depthM.

Corollary 3.5. Let A be a local ring, and let M be an A-module of finite CI-dimension and complexityc. Furthermore, let N be an A-module of finite length. If there exists a number n > depthA−depthM such that ExtiA(M, N) = 0 for nin+ c−1, then ExtiA(M, N) = 0 for all i >depthA−depthM.

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REFERENCES

1. Auslander,M.,Modules over unramified regular local rings, Illinois. J. Math. 5 (1961), 631–

647.

2. Avramov, L., Gasharov, V., Peeva, I.,Complete intersection dimension, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 67–114.

3. Avramov, L., Sun, L.-C.,Cohomology operators defined by a deformation, J. Algebra 204 (1998), 684–710.

4. Bergh,P. A.,On the vanishing of (co)homology over local rings, J. Pure Appl. Algebra 212 (2008), 262–270.

5. Bergh,P. A., Jorgensen, D.,On the vanishing of homology for modules of finite complete intersection dimension, J. Pure Appl. Algebra 215 (2011), 242–252.

6. Gulliksen, T. H.,A change of ring theorem with applications to Poincaré series and intersec- tion multiplicity, Math. Scand. 34 (1974), 167–183.

7. Gulliksen, T. H.,On the deviations of a local ring, Math. Scand. 47 (1980), 5–20.

8. Jorgensen, D.,Complexity andToron a complete intersection, J. Algebra 211 (1999), 578–

598.

9. Jorgensen, D.Vanishing of (co)homology over commutative rings, Comm. Alg. 29 (2001), 1883–1898.

10. Lichtenbaum, S.,On the vanishing ofTorin regular local rings, Illinois J. Math. 10 (1966), 220–226.

11. Murthy, M.,Modules over regular local rings, Illinois J. Math. 7 (1963), 558–565.

INSTITUTT FOR MATEMATISKE FAG NTNU

N-7491 TRONDHEIM NORWAY

E-mail:bergh@math.ntnu.no

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