Homological Algebra Modulo a Regular Sequence with Special … › bitstream › 1807 › 9851 › 1 › ... · 2007-07-26 · homological algebra modulo a regular sequence 25 CONTENTS

Journal of Algebra 230, 24–67 (2000)doi:10.1006/jabr.1999.7953, available online at http://www.idealibrary.com on

Homological Algebra Modulo a Regular Sequencewith Special Attention to Codimension Two

Luchezar L. Avramov1

Department of Mathematics, Purdue University, West Lafayette, IN 47907E-mail: [email protected]

and

Ragnar-Olaf Buchweitz2

Department of Mathematics, University of Toronto, Toronto,Ontario M5S 3G3, Canada

E-mail: [email protected]

Communicated by Craig Huneke

Received June 4, 1998

dedicated to david buchsbaum

Let M be a finite module over a ring R obtained from a commutative ring Q byfactoring out an ideal generated by a regular sequence. The homological propertiesM over R and over Q are intimately related. Their links are analyzed here fromthe point of view of differential graded homological algebra over a Koszul complexthat resolves R over Q. One outcome of this approach is a transparent derivation ofsome central results of the theory. Another is a new insight into codimension twophenomena, yielding an explicit finitistic construction of the generally infinite mini-mal R-free resolution of M. It leads to theorems on the structure and classificationof finite modules over codimension two local complete intersections that are exactcounterparts of Eisenbud’s results for modules over hypersurfaces. © 2000 Academic

Press

1 Partly supported by a grant from the NSF.2 Partly supported by a grant from the NSERC.

24

0021-8693/00 $35.00Copyright © 2000 by Academic PressAll rights of reproduction in any form reserved.

homological algebra modulo a regular sequence 25

CONTENTS

Introduction1. Homology operators2. Universal resolutions3. Cohomology operators4. Codimension two5. Minimal resolutions6. Spectral sequence7. Finite CI-dimension

INTRODUCTION

Regular sequences appeared simultaneously in two landmark papers pub-lished in 1956 at the outset of the “homological era” [24] of commutativering theory.

Auslander and Buchsbaum introduced regular sequences in [1] under thepresent name and used them to prove that regular local rings have finiteglobal dimension; details were given in [2]. Without naming them Rees [29]put regular sequences at the foundation of his theory of grade for idealsand its extension to modules in [30]. Since then, a number of authors havereturned to the general “change-of-rings” question:

How is the module theory of a commutative ring Q related to that of itsresidue ring R = Q/� f � when f = f1; : : : ; fc is a Q-regular sequence?

To put our results in context we provide a short historical overview.Rees [29] provided part of the answer: If an R-module M lifts to Q, in the

sense that M ∼= L/� f �L for a Q-module L on which f is regular, then thehomological properties of M over R are determined by those of L over Q.

Tate in 1957 constructed an R-free resolution F of Q/�g� when g is a Q-regular sequence and f ⊂ �g�. His paper [33] introduced to commutativealgebra the technique of DG (= differential graded) algebra resolutions, aversion of the “constructions” developed by H. Cartan for his celebratedcomputation of the homology of Eilenberg–MacLane spaces [15]. WhenR is a local complete intersection, meaning that Q can be taken to be aregular local ring, Tate’s construction yields a minimal free resolution ofthe residue field k of R.

In 1969 Shamash [31] significantly extended Tate’s result in the case ofcodimension 1, that is, when the regular sequence consists of a single ele-ment. Starting from a Q-free resolution E of an R-module M, he producedinductively a system σ of “higher homotopies” on E, fitted them togetherto get a differential on a free graded module with Fn =

⊕u¾0R⊗Q En−2u,

and showed that the resulting complex E�²� is a free resolution of M. As

26 avramov and buchweitz

a consequence, the Betti numbers βRn �M� = rankk TorRn �M;k� of any finitemodule M over a local complete intersection R are bounded above by apolynomial in n. If d − 1 is the minimal degree of such a polynomial M is(nowadays) said to have complexity d, denoted cxR M = d.

Gulliksen brought cohomology into the picture in 1974. The new toolin [21] is an action of a graded polynomial algebra S = R�χ1; : : : ; χc� onExt∗R�M;N� and on TorR∗ �M;N�, natural in both M and N. His operatorsχj of cohomological degree 2 are defined in terms of connecting homo-morphisms. When rings and modules are noetherian and the projectivedimension of M over Q is finite Ext∗R�M;N� is a noetherian S -module andTorR∗ �M;N� is a (relatively) artinian one. This result opened the road fromcommutative to homological algebra to two-way traffic.

In 1980, Eisenbud [18] took that road to study the case when Q is a regu-lar local ring. He proved that if cxR M ≤ 1 then the minimal free resolutionof M becomes periodic of period 2 after at most (depth R + 1) steps; incodimension 1 he expressed the periodic tail from a “matrix factorization”of the defining equation f . His key instruments were S -module structureson (co)homology induced by chain maps of degree −2 that operate on anyfree resolution of M and commute up to homotopy. Extending Shamash’sconstruction to arbitrary codimension, he produced a not necessarily mini-mal R-free resolution E�²� of M that is a DG module over S .

More recently, results on modules over complete intersections have ledto the introduction of new homological dimensions for modules over com-mutative rings. In [5, 9] most results on modules over complete intersec-tions have been extended to modules of finite CI-dimension over arbitrarylocal rings. The Betti sequence of such modules has been shown to growasymptotically as a polynomial in n. When cxR M ≤ 1, the sequence stabi-lizes after at most depth R+ 1 steps; when cxR M ≥ 2, it eventually strictlyincreases.

Next we describe the material in this paper.In Section 1 we recall a few basic facts from DG homological algebra

over the Koszul complex K resolving R over Q; complete details may befound in [7, Sect. 1]. The upshot is that if R ← Q → k are commutativering homomorphisms, then Ext∗Q�M;k� is naturally a graded module overTorQ∗ �R;k�. This is the cohomological counterpart of the classical homo-logical product of Cartan and Eilenberg [16].

Starting from a Q-free resolution E of M with action of the Koszul com-plex K� f yQ�, we produce in Section 2 a universal R-free resolution G�E�of M. It seems to be simpler to construct and to analyze than Tate’s reso-lution, which it generalizes, or that of Shamash and Eisenbud, of which itis a special case. Its explicit structure of DG module over S is used in Sec-tion 3 to introduce the operators in cohomology and to compute Exts inseveral cases of interest.


To describe the results of Section 4 we assume for simplicity that Qis a regular local ring with algebraically closed residue field k and M isa maximal Cohen–Macaulay module over a complete intersection R =Q/�f1; f2� of codimension 2. The resolution G�E� yields Ext∗R�M;k� asthe homology of a natural linear complex C•�M;k� of length two whoseterms are free graded modules over the polynomial ring R = k�χ1; χ2�.That complex is obtained by a standard procedure from the graded mod-ule M = Ext∗Q�M;k� over the exterior algebra T = TorQ∗ �R;k� on twovariables of degree 1. The indecomposable direct summands of M aredetermined by Kronecker’s theory of pencils of matrices and provide a de-composition of C•�M;k� into a direct sum of explicitly known linear com-plexes. All this allows description in detail of the R-module Ext¾nR �M;k�for n > m = 2 max�βR0 �M�; βR1 �M��.

Using the computation above, we trim in Section 5 the universal resolu-tion of the mth syzygy of M in order to get a minimal R-free resolution Fof M. The process shows that the truncation F>m has a structure of DGmodule over the ring of operators S = R�χ1; χ2�. This proves in codimen-sion 2 a conjecture of Eisenbud [18] that remains completely open in highercodimensions.

In Section 6 we return to a general ring Q and its residue ring R moduloa regular sequence f = f1; : : : ; fc . We set up a change-of-rings spectralsequence converging to Ext∗R�M;N�, when M and N are modules overR. Its second page is a linear complex C•�M;N� of graded S -modules,determined by the graded module Ext∗Q�M;N� over the exterior algebraR�ξ1; : : : ; ξc� = TorQ∗ �R;R�. The sequence yields a short transparent proofof Gulliksen’s finiteness theorem.

In the final Section 7 we focus on a module M of finite CI-dimensionover a local ring. From the spectral sequence we derive a new proof thatthe Betti sequence of M grows asymptotically like a polynomial. From theresults in codimension 2 we obtain a realistic estimate of that place in theresolution of a module of complexity 2 beyond which the Betti numbersstart to increase.

1. HOMOLOGY OPERATORS

In this section we prove the following two results.

1.1. Theorem. If R ← Q → k are homomorphisms of commutativerings, then T = TorQ∗ �R;k� is a graded associative and commutative alge-bra through pairings

TorQi �R;k� ⊗Q TorQj �R;k� → TorQi+j�R;k�that extend the canonical product on TorQ0 �R;k� = R⊗Q k.


For any R-module M and any k-module N there are pairings

TorQi �R;k� ⊗Q TorQj �M;N�→TorQi+j�M;N�;TorQi �R;k� ⊗Q ExtjQ�M;N�→Extj−iQ �M;N�

that endow TorQ∗ �M;N� and Ext∗Q�M;N� with a structure of graded mod-ule over T, extending the standard actions of R ⊗Q k on M ⊗Q N and onHomQ�M;N�.

These actions of T are natural in M and N; exact sequences of R-modulesor exact sequences of k-modules induce T-linear connecting morphisms.

Each k-module N ′ defines a natural morphism of graded TorQ∗ �R;k�-modules

Ext∗Q�M;Homk�N;N ′�� → Homk�TorQ∗ �M;N�;N ′�that is an isomorphism whenever the k-module N ′ is injective.

The algebra and module structures on Tor’s are classical, given by thehomology product ô of Cartan and Eilenberg [16].

As a first application of the theorem we show that the action of Tordetects free summands in certain cases. The largest rank of a free directsummand of a module L over a ring A is known as the free rank of L,denoted f-rankA L.

Recall that a finite module M 6= 0 over a local (noetherian) ring Q is saidto be perfect if its projective dimension proj dimQM equals the length of amaximal Q-regular sequence contained in its annihilator. An ideal Á ⊂ Q isGorenstein of codimension c if Q/Á is a perfect Q-module whose minimalfree resolution P has length c and satisfies Pc ∼= Q.

1.2. Proposition. Let Á be a Gorenstein ideal of codimension c in a localring Q with residue field k; set R = Q/Á and T = TorQ∗ �R;k�. If M is a finiteR-module with proj dimQM <∞ and depthM = depth R, then

f-rankRM = f-rankT Ext∗Q�M;k� = f-rankT TorQ∗ �M;k�:The notation of Theorem 1.1 remains in force throughout the section.

We describe next some basic DG algebra techniques used in various proofsof the paper.

1.3

The components of a graded Q-module L are indexed either by homolog-ical degree, written as a subscript, or by cohomological degree, appearingas a superscript. These degrees are interchangeable by the rule Li = L−ifor each i ∈ �; for u ∈ Li or u ∈ Li, we set �u� = �i�. If L and L′ are graded(left) modules over a graded algebra A, then a homomorphism λx L→ L′


of homological degree d is a family of additive maps λix Li → L′i+d fori ∈ �, such that λ�au� = �−1�diaλ�u� for all u ∈ L, a ∈ Ai, and i ∈ �;note that λ has cohomological degree −d. Homomorphisms of graded A-modules, of arbitrary degree, are also referred to as A-linear maps; degreezero homomorphisms are called morphisms.

1.4

For each s ∈ � we denote L�s� the graded abelian group with L�s�i =Li−s, or, equivalently, L�s�i = Li+s for all i ∈ �. Let 6sx L→ L�s� be themap L → L�s� of homological degree s defined by 6s�u� = u for all u,and turn L�s� into a graded A-module by setting a6s�u� = �−1�si6s�au�for a ∈ Ai and u ∈ L. In this way 6s becomes a degree s homomorphismof left graded A-modules.

For instance, if 0 → M ′ → M → M ′′ → 0 is an exact sequence of Q-modules, then the connecting maps ∂nx ExtnQ�M ′;N� → Extn+1

Q �M ′′;N� de-fine a connecting morphism of graded Q-modules ∂x Ext∗Q�M ′;N��−1� →Ext∗Q�M ′′;N�. Similarly, an exact sequence 0 → N ′ → N → N ′′ → 0 de-fines a connecting morphism ∂x Ext∗Q�M;N ′′��−1� → Ext∗Q�M;N ′�.

1.5

The graded module underlying a complex U is denoted U\. A morphismof complexes αx U → U′ is a morphism of graded Q-modules that satis-fies α∂ = ∂α. A quasi-isomorphism is a morphism such that Hi�α� is anisomorphism for all i.

A DG algebra K is a complex whose underlying graded module is agraded algebra and whose differential satisfies the Leibniz rule ∂�ab� =∂�a�b + �−1�ia∂�b� for a ∈ Ki and b ∈ K. Similarly, a DG module U overK is a complex such that U\ is a graded K\-module and ∂�au� = ∂�a�u +�−1�ia∂�u� for a ∈ Ki and u ∈ U.

Any ring can and will be considered a DG algebra concentrated in degreezero.

1.6. Example. Let f = �f1; : : : ; fc� be a set of elements of Q.The Koszul complex K� f yQ� is the DG algebra with K� f yQ�\ the exte-

rior algebra on a free Q-module with basis ξ1; : : : ; ξc of homological degree1, and differential ∂ such that ∂�ξj� = fj for j = 1; : : : ; c. It can also be de-scribed as the DG algebra obtained from Q by adjoining exterior variablesξ1; : : : ; ξc to kill the cycles f1; : : : ; fc; cf. [7, Sect. 6.1, 33]. One then writesK� f yQ� = Q�ξ1; : : : ; ξs � ∂�ξj� = fj�.

We say that a complex of Q-modules E has a Koszul structure if it is aDG module over K� f yQ�. By the Leibniz rule ∂�ξje� + ξj∂�e� = fje for alle ∈ E, so the map σ�j��e� = ξje of degree 1 is a Q-linear homotopy from


fj idE to 0E. Thus, a Koszul structure on E is nothing but a choice of afamily of homotopies

(σ�j�x fj idE ∼ 0E

)c1 satisfying σ�j�σ�i� = −σ�i�σ�j� and

σ�j�σ�j� = 0.

1.7

A DG module E with Ei = 0 for i � 0 is semi-free if the K\-module E\

is free. This is a special case of the notion in [10, Sect. 1], where generalfacts on semi-freeness are discussed. For elementary proofs of the followingpropositions cf. [7, Sect. 1.3].

Let E be a semi-free DG module and αx U′ → U a morphism of DGmodules.

1.7.1. If α is a surjective quasi-isomorphism of DG modules over K,then each morphism βx E → U lifts to a morphism β′x E → U′ such thatβ = αβ′; any two such liftings are homotopic by a K-linear homotopy.

1.7.2. If α is a quasi-isomorphism then so is α ⊗K Ex U′ ⊗K E →U⊗K E.

1.7.3. If E′ also is a semi-free DG module over K and if γx E′ → E isa quasi-isomorphism, then so is V ⊗K γx V ⊗K E′ → V ⊗K E for each DGmodule V.

Next we recall some standard constructions; details may be found in [7,Sect. 2].

1.8

Any ring homomorphism Q → R can be factored as a composition ofmorphisms of DG algebras Q ↪→ K � R with the following properties:The graded algebra K\ is the tensor product of the symmetric algebraon a free graded Q-module concentrated in non-negative even degreeswith the exterior algebra on a free graded Q-module concentrated in odddegrees; the surjection κx K � R is a quasi-isomorphism. If K′ → R isa quasi-isomorphism of DG algebras over Q, then there is a morphismK→ K′ of DG algebras over Q inducing the identity on R.

Any R-module M becomes a DG module over K through the morphismκ, and there is always a quasi-isomorphism E→ M from a non-negativelygraded semi-free DG-module E over K, cf. Subsection 2.1 for a specialcase.

1.9

Let K and E be as in Subsection 1.8. The complex HomQ �E;N� hasan induced structure of DG module over the DG algebra Λ = K ⊗Q k,


described by (�ξ⊗ s��ν�)�e� = �−1��ν�sν�ξe�for ξ ∈ K, e ∈ E, s ∈ k, and νx E → N, a homomorphism of gradedmodules.

Let K′ be a DG algebra with K′\ free over Q and H�K′� = R. If φx K→K′ is a morphism of DG algebras as in Subsection 1.8, then φ⊗K Ex E =K ⊗K E → K′ ⊗K E is a quasi-isomorphism by Proposition 1.7; clearly,K′ ⊗K E is a semi-free DG module over K′.

Let E′ be a DG module over K′ with H�E′� = M and with E′\ freeover Q. Proposition 1.7 yields a morphism βx K′ ⊗K E→ E′ of DG mod-ules over K′ that induces the identity on M. We get quasi-isomorphisms ofDG modules

HomQ�E′;N� HomQ�K′ ⊗K E;N�

HomQ�E;N�

HomQ�β;N�

HomQ�φ⊗KE;N�

with Λ acting on the first two via the morphism φ⊗Q kx Λ = K ⊗Q k→K′ ⊗Q k. Also, Proposition 1.7 shows that φ is unique up to K-linear ho-motopy and that β is unique up to K′-linear homotopy, so the compositonabove is unique up to Λ-linear homotopy.

1.10

Let µx M ′ → M be a homomorphism of R-modules. If E′ is a semi-free DG module over K′ with H�E′� = M ′, then Proposition 1.7 yields alifting of µ to a morphism of DG modules µx E′ → E. It is unique up tohomotopy, hence

HomQ�µ;N�x HomQ�E;N� → HomQ�E′;N�is a morphism of DG modules over Λ defined uniquely up to Λ-linearhomotopy.

Let 0 → M ′ → M → M ′′ → 0 be an exact sequence of R-modules.Choose resolutions E′ of M ′ and E′′ of M ′′ that are semi-free DG modulesover K. There exists a differential on E\ = E′\ ⊕ E′′\, turning it into aDG module over K such that H∗�E� = M and 0 → E′ → E → E′′ → 0becomes an exact sequence of DG modules. It splits over Q, inducing anexact sequence of DG modules over Λ

0→ HomQ�E′′;N� → HomQ�E;N� → HomQ�E′;N� → 0:


In particular, the associated connecting morphism in cohomology is Λ-linear.

We are ready to establish the results stated at the beginning of this sec-tion.

Proof of Theorem 1.1. Take K and E as in Subsection 1.8. As Kand E are Q-free resolutions of R and of M, respectively, we haveTorQ∗ �R;k� = H∗�K⊗Q k�, TorQ∗ �M;N� = H∗�E⊗Q N�, and Ext∗Q�M;N� =H∗�HomQ�E;N��. These expressions endow TorQ∗ �R;k� with a structureof graded algebra and TorQ∗ �M;N�, Ext∗Q�M;N� with that of graded mod-ule over it. By Subsection 1.9 these structures do not depend on thechoices of K and E. By Subsection 1.10 they are functorial in the first ar-gument, and connecting morphisms induced by exact sequences in thatargument are Λ-linear. It is clear that H∗�E⊗Q N� and H∗�E⊗Q N� havethe corresponding properties with respect to N.

By adjunction, H∗HomQ

(E;Homk�N;N ′�

) ∼= H∗Homk�E ⊗Q N;N ′�.Composed with the K’unneth map H∗Homk�E ⊗Q N; N ′� → Homk

�H∗�E⊗Q N�;N ′�, this isomorphism yields a canonical morphism

Ext∗Q�M;Homk�N;N ′�� → Homk�TorQ∗ �M;N�;N ′�of graded modules over TorQ∗ �R;k� that is bijective when N ′ is injective.

Proof of Proposition 1.2. Recall that Á is a codimension c Gorensteinideal in a local ring �Q;Î; k� and R = Q/Á. By [13, §1] the finite-dimensional k-algebra T = TorQ∗ �R;k� is self-injective. It follows that theisomorphism of T-modules Ext∗Q�M;k� ∼= Homk�TorQ∗ �M;k�; k� given byTheorem 1.1 transforms free direct summands into free direct summands,so f-rankT Ext∗Q�M;k� = f-rankT TorQ∗ �M;k�.

Choose a non-zero element ω ∈ Tc; note that it generates the socle of Tand set p = rankk�ω ·TorQ0 �M;k��. An R-free direct summand of M yieldsa T-free direct summand of TorQ∗ �M;k� of the same rank, hence

p ≥ f-rankT TorQ∗ �M;k� ≥ f-rankRM:

To finish the proof we show that f-rankRM ≥ p.Choose m1; : : : ;mp in M so that ω · �m1 ⊗ 1�; : : : ; ω · �mp ⊗ 1� is a basis

of ω · TorQ0 �M;k�. Let µx Rp → M be the homomorphism of Q-modulesdefined on the canonical basis by µ�ei� = mi. The induced map

TorQ∗ �µ;k�x TorQ∗ �Rp; k� → TorQ∗ �M;k�is T-linear, so TorQc �µ;k�

(ω · �ei ⊗ 1�) = ω · �mi ⊗ 1�; hence TorQc �µ;k� is

injective.


Let P be a minimal Q-free resolution of R and E be a minimal Q-freeresolution of M, and let µ′x Pp→ E be a morphism lifting µ. By Auslander–Buchsbaum

proj dimQM = depthQ− depthQM = depthQ− depthQ R

= proj dimQR = c;so Ec+1 = 0. Since µ′c ⊗Q k = TorQc �µ;k� is injective, µ′cx �Pc�p → Ec is amonomorphism onto a direct summand, and so the R-linear map

ExtcQ�µ;Q�x ExtcQ�M;Q� → ExtcQ�Rp;Q�is surjective. Setting �−�∨ = HomR�−;R� and recalling that ExtcQ�−;Q� ∼=�−�∨ as functors on the category of R-modules, we see that µ∨x M∨ →�Rp�∨ is surjective and hence split.

Dualizing once more we get a split monomorphism µ∨∨x Rp = �Rp�∨∨ →M∨∨. It is the composition of µx Rp → M with the biduality map M →M∨∨, so µ is a split monomorphism, and hence f-rankRM ≥ p, as desired.

2. UNIVERSAL RESOLUTIONS

In this section f = �f1; : : : ; fc� is a Koszul-regular subset of Q, in thesense that R = Q/� f � 6= 0� and the Koszul complex K = K� f ;Q� satisfiesHi�K� = 0 for i 6= 0; equivalently, the canonical projection κx K → R is anon-zero quasi-isomorphism.

2.1

Each R-module M has a Koszul resolution, meaning a projective resolu-tion of M over Q that is a DG module over K. In view of the importantrole such a resolution plays in the paper, we reproduce here its simpleconstruction.

Choose a surjection εx E =⊕i∈I�0�Rei�0� →M and let E�0� be the DGmodule with E�0�\ = ⊕i∈I�0�K

\ei�0� and ∂(ei�0�

) = 0. Let ε�0�x E�0� →M be the composition of ε with the obvious surjection E�0� → E. Forn ≥ 0, assume by induction that there is a DG module with E�n�\ =⊕

i∈I�n�K\ei�n� and a morphism ε�n�x E�n� → M with Hj�ε�n�� bijective

for j < n and surjective for j = n.Let �zi�n� ∈ E�n� � i ∈ In+1� be a set of cycles whose classes span

Ker Hn�ε�n��. Set I�n+ 1� = I�n� t In+1, and let E�n+ 1� be the DG mod-ule with E�n+ 1�\ =⊕i∈I�n+1�K

\ei�n+ 1� and differential that extends theone on E�n� and satisfies ∂

(ei�n+ 1�) = zi�n�. The unique K-linear map


ε�n+ 1�x E�n+ 1� → M that extends ε�n� and sends each ei�n+ 1� to 0is a morphism of DG modules over K. It is clear that Hε�n+ 1� is bi-jective in degrees < n + 1 and surjective in degree n + 1. The limit mapε = ⋃n¾0 ε�n� is then a quasi-isomorphism E → M and the DG moduleE = ⋃n¾0 E�n� is semi-free over K.

Note that En 6= 0 for n ≥ 0 if proj dimRM = ∞. However, ifproj dimQM = q < ∞, then setting E′n = 0 for n < q, E′q = ∂�Eq+1�and E′n = En for n > q defines a subcomplex E′ of E, and E′′ = E/E′ is aprojective resolution of M over Q. For degree reasons E′ is a DG submod-ule of E over Q, whence E′′ constitutes a Koszul resolution of M over Qthat has minimal possible length, equal to proj dimQM.

2.2

The graded algebra Λ = TorQ∗ �R;R� is computed as

Λ = H∗(Q�ξ1; : : : ; ξc � ∂�ξj� = fj� ⊗Q R

)= H∗

(R�ξ1; : : : ; ξc � ∂�ξj� = 0�) = R�ξ1; : : : ; ξc�:

The action of Λ on Ext∗Q�M;N� can be computed not only from a Koszulresolution of M, but from any projective resolution E′ of M over Q. In fact,if σ�j�x E′ → E′ is a homotopy fj idE′ ∼ 0E′ and if αx E′ → N is a Q-linearhomomorphism of cohomological degree i such that α∂ = 0, then

ξj cls�α� = �−1�i cls�ασ�j�� ∈ H∗HomQ�E′;N� = Ext∗Q�M;N�:

2.3

Let Q�χ1; : : : ; χc� be a polynomial ring on variables of cohomologicaldegree 2. For H = �h1; : : : ; hc� ∈ �c , we set χH = χh1

1 · · ·χhcc and notethat χH has cohomological degree 2�h1 + · · · + hc�.

The basis of � = HomQ�Q�χ1; : : : ; χc�;Q�, dual to the basis �χH � H ∈�c� of Q�χ1; : : : ; χc�, is denoted �y�H� � H ∈ �c�; accordingly, y�H� hashomological degree 2�h1 + · · · + hc�. The induced action of Q�χ1; : : : ; χc�on � is described by

χjy�H� =

{y�h1;:::;hj−1;:::;hc� if hj > 00 if hj = 0.

We set � = R⊗Q �. Abusing notation we write y�H� instead of 1⊗Q y�H�;the formulas above then give � an action of S = R ⊗Q Q�χ1; : : : ; χc� =R�χ1; : : : ; χc�.


2.4. Theorem. If E is a Koszul resolution of M, then the formula

∂�g ⊗ e� = g⊗ ∂�e� +c∑j=1

χjg⊗ ξje

defines an R-linear homomorphism ∂x � ⊗Q E\ → � ⊗Q E\ of homologicaldegree −1 that satisfies ∂2 = 0. The resulting complex G�E� = ��⊗Q E\; ∂� isa free resolution of M over R and a DG module over S ⊗Q Λ for the obviousaction.

2.5

Shamash [31, Sect. 3] for c = 1 and Eisenbud [18, Sect. 7] for any c pro-duce an R-free resolution of an R-module M out of any Q-free resolutionE′ of M over Q. They proceed in two steps, first exhibiting on E′ a systemof higher homotopies ² = �σ�H�x E′ → E′ � H ∈ �c�, then using it to de-fine on E′�²�\ = � ⊗Q E′\ a differential that turns E′ into a DG moduleover S with H∗�E′�²�� ∼=M.

When E′ has a Koszul structure the first step is gratuitous (cf. [18, p. 56]):just take σ�0��x� = ∂�x�, σ�j��x� = ξjx for the jth unit vector �j� andσ�H��x� = 0 for �H� > 1. With this choice, E′�²� = G�E′�, but we give adirect construction.

The key ingredient is a specific resolution of K viewed as a (left) DGmodule over K⊗Q K by means of the multiplication morphism, a⊗ b 7→ abfor a; b ∈ K; this goes back to H. Cartan’s “small constructions” from [15].

2.6. Proposition. Set L\ = K\ ⊗Q �⊗Q K\. The K\ ⊗Q K\-linear homo-morphism ∂x L\→ L\ of homological degree −1 given by

∂(y�H�

) = c∑j=1

(1⊗ χjy�H� ⊗ ξj − ξj ⊗ χjy�H� ⊗ 1

)turns L = �L\; ∂� into a DG module over K ⊗Q K, and the map

εx L→ K with ε(a⊗ y�H� ⊗ b) = {ab if H = 0 ,

0 if H 6= 0

is a quasi-isomorphism of DG modules over K ⊗Q K.

Proof. A simple computation yields ∂2 = 0. In what follows, unadornedtensor products are over Q. Clearly, ε is a �K ⊗ K�-linear chain map. Toshow that it is a quasi-isomorphism we use two alternative descriptions. LetL′ be the DG module over K ⊗K with L′\ = �⊗K\ ⊗K\ and differential

∂(y�H�

) = c∑j=1

(χjy�H�)⊗ ξ′j ; ξ′j = 1⊗ ξj − ξj ⊗ 1:


For j = 1; : : : ; c let T�j� be the complex of free Q-modules

· · · → Qy�n�j ξ′j

∂2n+1→Qy�n�j∂2n−→Qy�n−1�

j ξ′j → · · · → Qξ′j∂1−→Qy�0�j → 0

∂2n+1(y�n�j ξ′j

) = 0 and ∂2n(y�n�j

) = y�n−1�j ξ′j for n ≥ 1:

The canonical augmentations ε�j�x T�j� → H0�T�j�� = Q are clearlyquasi-isomorphisms. They appear in the commutative diagram(⊗c

j=1 T�j�)⊗K L′ L′

Q⊗c ⊗K K K

β α

(⊗cj=1 ε�j�

)⊗K

∼=

ε′ ε

where α�y�H� ⊗ a⊗ b� = a⊗ y�H� ⊗ b; ε′ = εα. For H = �h1; : : : ; hc� andij ∈ �0; 1�,

β(y�h1�1 ξ′i11 ⊗ · · · ⊗ y�hc�c ξ′icc ⊗ b

) = y�H� ⊗ ξ′1i1 · · · ξ′cic b:The map α is obviously an isomorphism.The map β is bijective, because it is linear for the actions of K on the

rightmost factors and maps a K\-basis of(⊗c

j=1 T�j�\)⊗K\ to one of L′\.The map

(⊗cj=1 ε�j�

) ⊗ K is a quasi-isomorphism by the Kunneththeorem.

The commutativity of the diagram implies that ε is a quasi-isomorphism.

Proof of Theorem. 2.4. In the notation of the preceding proof, the maps

R⊗K L ⊗K E←κ⊗KL⊗K E K ⊗K L ⊗K E ∼= L ⊗K E ε⊗KE→K ⊗K E ∼= E

are morphisms of complexes of Q-modules.For the action via the right-hand factor K in K ⊗K ⊆ L, the DG mod-

ule L is semi-free over K. Thus, ε is a quasi-isomorphism of semi-freeK-modules and Proposition 1.7 shows that ε⊗K E is a quasi-isomorphism.

The DG module L ⊗K E over K is semi-free for the action via the left-hand factor K in K ⊗K ⊆ L, because �L⊗K E�\ ∼= K\ ⊗ �⊗ E\ and the Q-modules � and E\ are free. As κ is a quasi-isomorphism, so is κ⊗K L⊗K Eby Proposition 1.7.

Via these quasi-isomorphisms we get H�R⊗K L ⊗K E� ∼= H�E� = M. Itremains to note that the isomorphism of graded modules

�⊗Q E\ ∼= �R⊗K K ⊗ �⊗K ⊗K E�\ ∼= �R⊗K L ⊗K E�\

induces an isomorphism G�E� ∼= R⊗K L⊗K E of DG module over S ⊗Q Λ.


The situation in codimension 1 was investigated further by Eisenbud [18].We present some of his results from the point of view of Theorem 2.4 thatbecomes particularly simple in this case. An alternative approach is takenby Iyengar [23, 3.2], who recovered the periodic resolution from a universalresolution based on the bar-construction.

2.7. Example. Let f be a Q-regular element, R = Q/�f �, and let M bea finite R-module. Assume first that M has a Q-free resolution of the form

E = 0→ E1δ→E0 → 0:

As fM = 0, we have fE0 ⊆ δ�E1�, so there is an R-linear map σ0x E0 → E1with δσ0 = f idE0 . Because δσ0δ = f idE0 δ = δf idE1 and δ is injective, wealso have σ0δ = f idE1 . Thus, σ0 defines a homotopy σ from f idE to 0E.It trivially satisfies σ2 = 0, so ξx = σ�x� turns E into a DG module overK = Q�ξ �δ�ξ� = f � by Subsection 3.4. In positive degrees 2n+ 2, 2n+ 1,and 2n, the resolution G�E� takes the form

· · · R⊗Qδ−→Ry�n+1� ⊗Q E0R⊗σ−→Ry�n� ⊗Q E1

R⊗δ−→Ry�n� ⊗Q E0R⊗σ−→· · · :

The graded algebra R�χ��ξ� operates on G�E� by χ�y�n� ⊗ x� = y�n−1� ⊗ xand ξ�y�n� ⊗ x� = y�n� ⊗ σ�x�.

When Q is local and E is a minimal Q-free resolution we have Im�R⊗δ� ⊆ ÍRy�n� ⊗Q E0. Thus, G�E� is minimal precisely when Imσ ⊆ ÎE1. Byexactness and periodicity we have Im�R ⊗ δ� = Coker�R ⊗Q δ� ∼= M, soG�E� is minimal if and only if M has no free direct summand.

Keeping Q local, consider more generally an R-module M withproj dimQM < ∞, and let F be its minimal free resolution over R. Ifdepth R − depthR M = g and M ′ is an nth syzygy of M with n ≥ g, thendepthR M

′ = depth R; if n > g, then in addition M ′ has no free directsummand, cf. [18, 7, (1.2.5)], so F>n ∼= G�E′�, where E′ is a minimal freeresolution of M ′ over Q.

Another source of free resolutions with Koszul structure is the following.

2.8

If ϕx Q→M is a homomorphism of Q-algebras with ϕ� f � = 0 and E isa resolution of M over Q that is a strictly commutative DG algebra, then Ehas a Koszul structure. Indeed, choose ζ1; : : : ; ζc ∈ E1 with ϕ�fj� = ∂�ζj�for j = 1; : : : ; c. By the universal property of the exterior algebra ϕ extendsto a morphism of graded Q-algebras K\ → E\ that maps ξj to ζj for j =1; : : : ; c. This map commutes with the differentials on the generators ofK, thus defining a morphism of DG algebras K → E and hence a Koszulstructure on E.


Here are some instances to which this observation applies.

2.8.1. Let M = Q/�g� where g = �g1; : : : ; gd� is a Q-regular set withf ⊂ �g�, say fj =

∑di=1 aijgi for j = 1; : : : ; c. If E = Q�ζ1; : : : ; ζd � ∂�ζi� =

gi� is the Koszul complex resolving M over Q, then by Subsection 2.8 thereis a unique morphism of DG algebras φx K→ E with φ�ξj� =

∑di=1 aijζi.

In this special case G�E� is Tate’s resolution of M over R, denoted

R

⟨ζ1; : : : ; ζdy y1; : : : ; yc

∣∣∣∣∂�ζi� = gi y ∂�yj� = d∑i=1

aijζi

⟩in [33, Theorem 4]. The proof of Theorem 2.4 does not use Tate’s result;one may as well reverse the roles, as was done in [7, (9.1.1)].

2.8.2. With f and g as above, assume that f ⊂ �g�h for some h ≥ 1.Buchsbaum and Eisenbud [14, (3.2)] construct a free resolution E of M =Q/�g�h over Q with ∂�E� ⊆ �g�E. The rank of Ei, computed from [14,(2.5.c)], is equal to

ai�h; d� =(h+ d − 1h+ i− 1

)(h+ i− 2h− 1

):

Srinivasan [32, (3.4), (3.6)] puts on E a DG algebra structure with E¾1 ·E¾1 ⊆ �g�E.

2.8.3. Buchsbaum and Eisenbud [13, (1.3)] show that if E0 = Q andEi = 0 for i > 3, then E has a structure of DG algebra.

We finish this section with a further remark on multiplicative structures.

2.9

Let � be the graded R-module from Subsection 2.3. The multiplicationtable

y�H�y�H′� =

(H +H ′H

)y�H+H

′� for H;H ′ ∈ �c

turns � into a graded R-algebra, known as the free divided powers algebraover R in the 0-variables y1; : : : ; yc . If E is a DG algebra over K, thenG�E� becomes a DG algebra with underlying graded algebra �⊗Q E\, andthe inclusion E = E⊗Q 1 ⊆ G�E� is one of DG algebras. In particular, theDG module L of Proposition 2.6 is a DG algebra that contains K ⊗Q K asa DG subalgebra.

In the parlance of Quillen’s homotopical algebra [27], transposed byhim into commutative algebra in [28], the DG algebra maps K ⊗ K →L → K constitute a model of the multiplication map K ⊗ K → K andS ∼= H�HomK⊗K�L;R�� represents the (derived or hyper-) Hochschild co-homology of the algebra Q→ R.


3. COHOMOLOGY OPERATORS

In this section f = �f1; : : : ; fc� is a Koszul-regular set in a commutativering Q, K is the Koszul complex on f , and R = Q/� f �.

Gulliksen [21] proves that Ext and Tor functors over R are naturallygraded modules over a polynomial ring with variables of cohomologicaldegree 2. Eisenbud [18] introduces another action and the two structuresagree up to sign; cf. [11].

We give a new construction of Eisenbud’s operators. It is based onresolutions of R-modules that are DG modules over the graded algebraS ⊗Q Λ, where the ring of cohomology operators S is the polynomial ringR�χ1; : : : ; χc� with variables χj of cohomological degree 2 and the algebraof homology operators Λ is the exterior algebra R�ξ1; : : : ; ξc� with variablesξj of cohomological degree −1.

3.1. Theorem. For R-modules M, N the algebra S acts on Ext∗R�M;N�through

χjx ExtnR�M;N� → Extn+2R �M;N� for n ∈ � and j = 1; : : : ; c:

The action is natural in both M and N.Exact sequences of R-modules induce S -linear connecting morphisms.

The main work is embedded in the following constructions.

3.2

To remind the reader that the variables χj have cohomological degree2 we use � � to denote shifts on graded S -modules. For instance, multi-plication with a linear form in χ1; : : : ; χc defines a morphism from S toS �2�.

For a Koszul resolution E of M let C�E;N� be the DG module over S ,with

C�E;N�\ =∞⊕u=0

S ⊗R HomQ�Eu;N��−2u�

and canonical action of S , whose differential acts by the formula

∂�χH ⊗ γ� = �−1��γ�(χH ⊗ γ ◦ ∂E +c∑j=1

χjχH ⊗ γ ◦ ξj

):

One sees by direct comparison that C�E;N� = HomR�G�E�;N� as com-plexes of R-modules, where G�E� is the R-free resolution of M providedby Theorem 2.4.


We bigrade the R-module underlying C�E;N� by assigning to the ele-ments of S v ⊗R Hom�Eu;N��−2u� complex degree −u, operator degree2u + 2v, and bidegree �−u; 2u+ 2v�; the sum of the complex and opera-tor degrees of a bihomogeneous element is its cohomological degree, hereequal to u+ 2v.

The action of χj ∈ S has bidegree �0; 2� and cohomological degree 2.The action of ξj ∈ Λ has bidegree �1;−2� and cohomological degree −1The first component of ∂ has bidegree �−1; 2� and cohomological de-

gree 1.The second component of ∂ has bidegree �1; 0� and cohomological de-

gree 1.Thus, C�E;N� is a DG module over the algebra S ⊗R Λ with polynomial

variables χj of bidegree �0; 2� and exterior variables ξj of bidegree �−1; 2�.As in Subsection 1.9, if E′ is any Koszul resolution of M, then C�E′;N� andC�E;N� are linked by quasi-isomorphisms of DG modules over S ⊗R Λdefined uniquely up to homotopy.

3.3

Let µx M ′ → M be a homomorphism of R-modules. If E′ is a Koszulresolution of M ′ and if µx E′ → E is a lifting of µ to a morphism ofDG modules over K, then S ⊗Q µx G�E′� → G�E� is a lifting of µ to amorphism of DG modules over S ⊗R Λ. If µ′x E′ → E also lifts µ, thenProposition 1.7 provides a K-linear homotopy τx µ ∼ µ′, so S ⊗Q τ is anS ⊗R Λ-linear homotopy S ⊗Q µ ∼ S ⊗Q µ′.

Let 0→ M ′ → M → M ′′ → 0 be an exact sequence of R-modules. BySubsection 1.10 it is the homology sequence of an exact sequence 0 →E′ → E → E′′ → 0 of DG modules over K that splits over Q. The lattersequence gives rise to the exact sequence of DG modules over S ⊗R Λ

0→ C�E′′;N� → C�E;N� → C�E′;N� → 0:

Proof of Theorem 3.1. Let S act on Ext∗R�M;N� through the isomor-phism

Ext∗R�M;N� ∼= H∗�C�E;N��

that results from Subsection 3.2. The discussion in Subsection 3.3 showsthat this structure is independent of the choice of E, that it is functorialin M, and that exact sequences in the first argument induce connectingmorphisms of S -modules.

The corresponding properties in the second argument are clear.


3.4

As Ext∗R�M;N� is the cohomology of a DG module over S ⊗Q Λ, italso inherits a structure of graded module over the algebra Λ of homologyoperators.

However, this action is trivial: ξj Ext∗R�M;N� = 0 for j = 1; : : : ; c.Indeed, let L be the DG module over K⊗Q K defined in Proposition 2.6.

Consider map τ′jx L→ L of degree 2 given by

τ′j(a⊗ y�h1; ::: ;hj; ::: ;hc� ⊗ b) = �hj + 1�a⊗ y�h1; ::: ;hj+1; ::: ;hc� ⊗ b:

A straightforward computation then shows that τ′j is a K ⊗Q K-linear ho-motopy between ξ′j idL and 0. It follows that it induces a homotopy betweenthe action of ξj and the zero map on E�ξ� = R⊗K L ⊗K E and hence be-tween the action of ξj and the zero map on C�E;N� = HomR�E�ξ�;N�.The desired assertion follows.

3.5

Theorem 3.1 gives rise to some useful canonical morphisms.

3.5.1. The homomorphism Q→ R induces a change-of-rings morphism

ρMN x Ext∗R�M;N� → Ext∗Q�M;N�:This is the map induced in homology by the morphism of complexes ofR-modules

C�E;N� → C�E;N��χ1; : : : ; χc�C�E;N�

∼= HomQ�E;N�:

It induces a decomposition of ρMN of the form

Ext∗R�M;N� →Ext∗R�M;N�

�χ1; : : : ; χc� Ext∗R�M;N�ρMN→Ext∗Q�M;N�

in which the first map is the canonical projection; we refer to the secondmap as the reduced change-of-rings morphism. By construction, ρMN is amorphism of graded Λ-modules, so the preceding Remark 3.4 implies that

Im�ρMN� ⊆ annExt∗Q�M;N��ξ1; : : : ; ξc�= �ε ∈ Ext∗Q�M;N� � ξjε = 0 for 1 ≤ j ≤ c�:

3.5.2. The action of S on Ext0R�M;N� is described by the morphism

~MN x S ⊗R Ext0R�M;N� → Exteven

R �M;N�of graded S -modules induced by the embedding of complexes

S ⊗R Ext0R�M;N� = S ⊗R HomQ�M;N� ⊆ S ⊗R HomQ�E;N�:


3.6

Let F be an R-projective resolution of M and M ′ = ∂�Fs�.The iterated connecting morphism ∂x Ext∗R�M ′;N��−s�→Ext∗R�M;N� is

surjective in degree s and bijective in degrees > s. When R is local, M isfinite, and F is its minimal resolution, the homomorphism ∂ is bijective indegrees ≥ s. As ∂ is S -linear, the S -module Ext can be studied by shiftingdegrees.

In this connection it is useful to note the following: Due to the iso-morphisms ExtiQ�Fj;R� ∼=

∧i�Rc� ⊗R HomR�Fj;R�, if ExtiQ�M;N� = 0 fori > g then ExtiQ�M ′;N� = 0 for i > max�c; g− s� by the cohomology exactsequence.

3.7

Assume that M has a Koszul resolution E over Q such that HomQ�∂E;N�= 0.

In this case the DG module C�E;N� simplifies to a complex C•�M;N� ofgraded S -modules with C−p�M;N� = S ⊗R Ext−pQ �M;N��2p� and whosedifferential d is given by multiplication with

∑cj=1 χj ⊗R ξj; note that d

has complex degree 1, operator degree 0, and cohomological degree 1.Accordingly, Ext∗R�M;N� acquires a canonical decomposition as a directsum of graded S -modules

ExtevenR �M;N�=

⊕p even

H−p�C•�M;N��p�;

ExtoddR �M;N�=

⊕p odd

H−p�C•�M;N��p�:

This computation applies, in particular, to the case when �Q;m;k� is alocal ring, M has a minimal free Koszul resolution E over Q, and N = k.

3.8. Example. Let P = k�x1; : : : ; xd� be a polynomial ring over a com-mutative ring k and let h = h1; : : : ; hc be a Koszul-regular sequence in Psuch that R = P/�h� is flat over k. In this context, overbars denote imagesin R.

The kernel of the multiplication map P ⊗k P → P is generated by theregular sequence dx = 1⊗x1−x1⊗ 1; : : : ; 1⊗xd −xd⊗ 1. Set Q = P⊗k Rand let g = g1; : : : ; gc be the image of dx in Q, so that gi = 1⊗ xi − xi ⊗ 1.Clearly,

Q/�g� ∼= P ⊗P⊗kP �P ⊗k R� ∼= R:The isomorphism K�gyQ� ∼= K�dxyP ⊗k P� ⊗P⊗kP Q yields

H∗�K�gyQ�� ∼= TorP⊗kP∗ �P;Q�:


By assumption, the Koszul complex K�hyP� is a free resolution of Rover P . This implies that P ⊗k K�hyP� is a free resolution of Q overP ⊗k P , so

TorP⊗kP∗ �P;Q� ∼= H∗(P ⊗P⊗kP �P ⊗k K�hyP��) ∼= H∗�K�hyP�� ∼= R:

These isomorphisms show that K = K�gyQ� is a resolution of R over Q.Setting f = f1; : : : ; fc with fj = hj ⊗ 1, we see that the complex E =

K� f yQ� is isomorphic to K�hyP� ⊗k R and hence resolves R⊗k R over Q,as R is k-flat.

Since Q/�g� ∼= R, we have f ⊆ �g�; hence HomQ�∂E;N� = 0 for eachR-module N. Now Subsection 3.7 yields a direct sum decomposition

ExtnR⊗kR�R;N� ∼=⊕

−p+q=nH−p�C•�M;N��q

along with a similar decomposition for TorR⊗kRn �R;N�. These Tor’s rep-resent Hochschild homology as R is k-flat; the preceding Ext’s representHochschild cohomology if R is furthermore projective as k-module.

If k contains �, then the decompositions above coincide with the Hodge(or λ-) decompositions of Hochschild (co-)homology; cf. [26, 4.5.10]. Theexistence of such a decomposition for complete intersections of arbitrarycharacteristic was apparently first recorded by J. A. Guccione and J. J.Guccione [20]. Furthermore, Subsection 2.8.1 provides an explicit resolutionof R = Q/�g� over R ⊗k R: it is the one constructed by Wolffhardt [34,Theorem 2] through a direct computation.

3.9. Example. Let �Q;Î; k� be a local ring; let g = g1; : : : ; gd and f =f1; : : : ; fc be Q-regular sequences with f ⊆ �g�h for some h ≥ 1. Set R =Q/� f � and M = Q/�g�h. Changing the generators of f if necessary, we mayfurther assume that f1; : : : ; fb project to a k-linearly independent subset of�g�h/Î�g�h, while fb+1; : : : ; fc are in Î�g�h. Resolve R over Q by the Koszulcomplex K on f .

3.9.1. If h = 1, changing if necessary the generators of g, we may as-sume that gj = fj for 1 ≤ j ≤ b. Resolve M by the Koszul complex Eon g. The morphism of DG algebras φx K → E from Subsection 2.8 in-duces a homomorphism of algebras φx k�ξ1; : : : ; ξc� → k�ζ1; : : : ; ζd� withφ�ξj� = ζj for j ≤ b and φ�ξj� = 0 for j > b. Thus, we get an isomorphismof complexes of graded R-modules

C•�M;k� ∼= K�χ1; : : : ; χb; 0; : : : ; 0yR�with C•�M;k� as in Subsection 3.7. It yields an isomorphism of gradedR-modules

Ext∗R�M;k� ∼= A⊗k∧∗(kd−b�−1�)


where the graded R-algebra A is defined by

A = R

�χ1; : : : ; χb�∼= k�χb+1; : : : ; χc�:

3.9.2. If h ≥ 2, let E be the minimal DG algebra resolution of M fromSubsection 2.8, and let φx K→ E be a morphism of DG algebras that liftsthe canonical projection R = Q/� f � → Q/�g�h =M. Since E¾1 ·E¾1 ⊆ ÎE,we get

TorQ∗ �M;k� =T

�ξb+1; : : : ; ξc�T + T¾2⊕ ka1�h; d�−b�−1�

d⊕i=2

kai�h; d��−i�

as DG modules over T = k�ξ1; : : : ; ξc�. From Theorem 1.1 we obtain anisomorphism Ext∗Q�M;k� ∼= Homk�TorQ∗ �M;k�; k� of graded T-modules,and so Subsection 3.7 shows that Ext∗R�M;k� can be computed from acomplex C�M;k� with Ci�M;k� = Rai�h; d��−2i� for i ≥ 0 and with onlynon-zero differential

Ra1�h; d��−2� �χ1 ···χb 0 ··· 0�→R:

Thus, we get an isomorphism of graded R-modules

Ext∗R�M;k� ∼= A ⊕B�1� ⊕Ra1�h; d�−b�−1�d⊕i=2

Rai�h; d��−i�

where ai�h; d� =(h+d−1h+i−1

)(h+i−2h−1

)as in Subsection 2.8.2, A is the R-algebra

defined in Subsection 3.9.1, and

B = Ker(RC�−2� �χ1···χb�→R

):

3.10

We recall the definitions of some invariants attached to a local ring�R;Í; k�.

The embedding dimension of R is the minimal number of generators ofÍ, denoted edimR. If edimR = d, then codimR = d − dimR is the codi-mension of R. We define the order of R by setting ordR = 1 if R is regularand ordR = inf�n ∈ � � lengthR�R/Ín+1� < (

d+nn

)� otherwise.For any minimal Cohen presentation of the completion R of R as a

residue Q/Á of a regular local ring �Q;Î; k� by an ideal Á ⊆ Î2 one hasedimR = dimQ, codimR = height Á, and ordR = sup�n ∈ � � Á ⊆ În�unless R is regular.

A local ring R is a complete intersection if the defining ideal in some (or,equivalently, in any) Cohen presentation R = Q/Á can be generated by a


regular sequence. We can determine the R-modules Ext∗R�R/Ín; k� for the

initial values of n by applying Subsection 3.9 to a regular sequence g ofgenerators of the maximal ideal of Q. The numerical expressions of thesecomputations are formulas for Poincare series recorded in the followingcorollary. For n = 1 the result goes back to Tate [33, Theorem 6]; for n = 2it is proved in [6, (2.1)] by a more complicated argument.

3.11. Corollary. Let �R;Í; k� be a local complete intersection with em-bedding dimension d, codimension c, and order h. If lengthR�R/Íh+1� =(d+hh

)− b, then

∞∑j=0

βRj �R/Ín�tj

=

∑di=0 ai�n; d�ti�1− t2�c if n < h,∑di=1 ai�n; d�ti�1− t2�c + �1+ t��1− t

2�b − 1t�1− t2�c if n = h.

The complex C•�M;k� appearing in Subsection 3.7 is closely related tothe construction introduced by J. Bernstein et al. in their study [12] of thederived category of the category of coherent sheaves on projective space;cf. the exposition of S. Gelfand [19] for details. That theory is not usedhere, so we only sketch the connection.

3.12

Let T = k�ξ1; : : : ; ξc� be the exterior algebra on variables of homolog-ical degree 1 and R = k�χ1; : : : ; χc� be the polynomial algebra on vari-ables of cohomological degree 2. We identify the k-vector spaces T1 andHomk�R2; k� by proclaiming ξ1; : : : ; ξc and χ1; : : : ; χc dual bases.

To a graded T-module M one associates the complex of graded R-modules

M• = · · · → R⊗k Mp�−2p� ∂→R⊗k Mp−1�−2p+ 2� → · · ·whose differential is given by multiplication with

∑cj=1 χj ⊗ ξj . The complex

M• is linear, in the sense that in terms of bases of the modules Mp thedifferential is given by matrices of linear forms in χ1; : : : ; χc; it is alsostandard, meaning that each Mp is generated in degree 2p.

In the opposite direction, for each standard linear complex of gradedR-modules

N • = · · · → N p+1 ∂→N p ∂→N p−1 → · · ·


let N be the graded vector space with Np = N p ⊗R k. The differential ∂induces k-linear maps Np → R2 ⊗k Np−1 for p ∈ �. Under the canonicalisomorphisms

Homk�Np;R2 ⊗k Np−1� ∼= Homk�Homk�R2; k� ⊗k Np;Np−1�= Homk�T1 ⊗k Np;Np−1�

these maps define k-linear homomorphisms T1⊗k Np→ Np−1. The equality∂2 = 0 guarantees that they produce on N a structure of graded T-module.

The assignments M 7→ M• and N • 7→ N are quasi-inverse additive equiva-lences between the category of graded T-modules and the category of stan-dard linear complexes of graded R-modules. This equivalence is describedby Yoshino in [35]. The functors themselves are restrictions of functors be-tween the categories of complexes of graded T-modules and complexes ofgraded S -modules; the equivalence above may be viewed as a specializa-tion of the BGG correspondence [12, 19].

4. CODIMENSION TWO

This section is devoted to the proof of the following result.

4.1. Theorem. Let R = Q/�f1; f2� where f1; f2 is a regular sequencein a local ring �Q;Î; k�, and let S = R�χ1; χ2� be the corresponding ringof cohomology operators. Let M be a finite R-module, and set depth R −depthR M = g.

If M has finite projective dimension over Q then Ext∗R�M;k� is generatedover the ring R = S ⊗R k = k�χ1; χ2� by elements of degree ≤ m, where

m = max{2βRg �M� ; 2βRg+1�M� + 1

}+ g + 1:

This is an effective version of a general finiteness result of Gulliksen [21];cf. Corollary 6.2 below. Our proof establishes much more than the state-ment above: in Subsection 4.7 we obtain Ext¾mR �M;k� as a direct sum ofexplicitly given indecomposable graded R-modules.

We do not know whether the bound on the degrees of the generators issharp. By applying an argument from the proof of [9, (7.4)] to an exampleof [18, p. 44] we see that it cannot be cut more than in half:

4.2. Example. Let R = k��x1; x2��/�f1; f2�, so that g = 0 for each finiteR-module. Tate’s minimal free resolution G of k over R (cf. Subsection 2.8).has rankRGn = n+ 1 for n ≥ 0. Since R is self-injective, dualization yieldsan exact sequence

0→k→ HomR�G0;R�→ · · · → HomR�Gs−1;R�∂−s−→HomR�Gs;R� → · · ·


with Ms = Im�∂−s� ⊆ m HomR�Gs;R�. The R-module Ms has βR0 �Ms� = sand βR1 �Ms� = s − 1; by the theorem, Ext∗R�Ms; k� is generated in degree≤ 2s + 1.

Since k is an sth syzygy of Ms, Subsection 3.6 yields an exact sequence

0→ Ext∗R�k;k��−s� → Ext∗R�Ms; k� → N → 0

of graded R-modules, with rankkN <∞. By Subsection 3.9 the R-moduleExt∗R�k;k� is free, with one basis element apiece in degrees 0 and 2 andtwo basis elements in degree 1. It follows that Ext1

R�N ;Ext∗R�k;k�� = 0, sothe sequence splits, showing that Ext∗R�Ms; k� has a minimal generator indegree s + 2.

We now take a careful look at graded modules over the algebraTorQ�R;k�.

4.3

Let T = k�ξ1; ξ2� be the exterior algebra on variables ξ1; ξ2 ∈ T1 = T−1.

4.3.1. As the algebra T is self-injective K = Homk�T; k� is a free T-module on a basis element contained in K−2 = K2. Each graded free T-module is isomorphic to a direct sum of translates of K. A T-module N hasa no free direct summand if and only if ξ1ξ2N = 0, that is, if and only if itis a module over the algebra T/�ξ1ξ2�.

A straightforward way to produce T/�ξ1ξ2�-modules is to fix an integerp, set Ni = 0 for i 6= p;p − 1, choose finite dimensional k-vector spacesNp and Np−1 and define the action of T/�ξ1ξ2� by specifying linear mapsξ1; ξ2x Np → Np−1. Any such choice yields a module structure. Here aresome simple examples.

4.3.2. For each integer n ≥ 0 let L�n� be the T-module with non-zerocomponents L�n�1 = kn and L�n�0 = kn+1 and action

ξ1 =

1 0 0 · · · 0 00 1 0 · · · 0 00 0 1 · · · 0 0:::

::::::: : :

::::::

0 0 0 · · · 1 00 0 0 · · · 0 10 0 0 · · · 0 0

ξ2 =

0 0 0 · · · 0 01 0 0 · · · 0 00 1 0 · · · 0 0:::

::::::: : :

::::::

0 0 0 · · · 0 00 0 0 · · · 1 00 0 0 · · · 0 1

:


For each integer n > 0 let L�n� be the T-module with non-zero compo-nents L�−n�0 = kn+1 and L�−n�−1 = kn and action

ξ1 =

1 0 0 · · · 0 0 00 1 0 · · · 0 0 00 0 1 · · · 0 0 0:::

::::::: : :

::::::

:::0 0 0 · · · 1 0 00 0 0 · · · 0 1 0

ξ2 =

0 1 0 · · · 0 0 00 0 1 · · · 0 0 00 0 0 · · · 0 0 0:::

::::::: : :

::::::

:::0 0 0 · · · 0 1 00 0 0 · · · 0 0 1

:

Clearly, L�0� is the residue field k = T/�ξ1; ξ2�. It is easy to see that forn > 0 the T-module L�−n��n� is a nth syzygy of k and is isomorphic toHomk

(L�n��−n�; k).

4.3.3. For each integer n > 0 and each λ ∈ k ∪ �∞� = �1k, let M�n; λ�

be the T-module with non-zero components M�n; λ�1 = kn and M�n; λ�0 =kn and action

ξ1 =

1 0 0 · · · 0 00 1 0 · · · 0 00 0 1 · · · 0 0:::

::::::: : :

::::::

0 0 0 · · · 1 00 0 0 · · · 0 1

ξ2 =

λ 0 0 · · · 0 01 λ 0 · · · 0 00 1 λ · · · 0 0:::

::::::

: : ::::

:::0 0 0 · · · λ 00 0 0 · · · 1 λ

if λ 6= ∞;

ξ1 =

0 0 0 · · · 0 01 0 0 · · · 0 00 1 0 · · · 0 0:::

::::::: : :

::::::

0 0 0 · · · 0 00 0 0 · · · 1 0

ξ2 =

1 0 0 · · · 0 00 1 0 · · · 0 00 0 1 · · · 0 0:::

::::::: : :

::::::

0 0 0 · · · 1 00 0 0 · · · 0 1

if λ =∞:

The significance of the modules just described is due to a classical resultof Kronecker [25]: For k = �, the pairs �ξ1; ξ2� appearing above forma complete list of indecomposable pairs of commuting complex matrices�ξ1; ξ2� up to similarity. Dieudonne [17] gave the first modern proof, overan arbitrary algebraically closed field. Heller and Reiner [22] reinterpretedthe result as a description of the isomorphism classes of indecomposable(not necessarily graded) modules. We abstract:


4.4

The T-modules K, L�n�, M�n; λ�, with n ∈ � and λ ∈ �1k, are indecom-

posable. When k is algebraically closed, each finite graded T-module is iso-morphic to a direct sum of shifts of these. Such a decomposition is uniquein the sense of Krull–Schmidt.

4.5

By Subsection 3.12, the classification of indecomposable graded T-modules yields a classification of the indecomposable linear complexesover R = k�χ1; χ2�.

4.5.1. The module K = Homk�T; k� corresponds to the Koszul complex

K = 0→ R�−4�(−χ2χ1

)→R2�−2� �χ1 χ2�→R→ 0:

4.5.2. For n ≥ 0 the module L�n� of Subsection 4.3.3 corresponds to thecomplex

L•�n� = 0→ Rn�−2� → Rn+1 → 0

concentrated in complex degrees −1 and 0, with non-zero differential

χ1 0 0 · · · 0 0χ2 χ1 0 · · · 0 00 χ2 χ1 · · · 0 0:::

::::::

: : ::::

:::0 0 0 · · · χ1 00 0 0 · · · χ2 χ10 0 0 · · · 0 χ2

:

Its only non-zero homology is in degree 0, where the Hilbert–Burch theo-rem yields H0�L•�n�� ∼= L�n�, with L�n� = �χ1; χ2�n�2n�.

For n > 0 the module L�−n� of Subsection 4.3 corresponds to the com-plex

L•�−n� = 0→ Rn+1 → Rn�2� → 0

concentrated in complex degrees 0 and 1, with non-zero differential

χ1 χ2 0 · · · 0 0 00 χ1 χ2 · · · 0 0 00 0 χ1 · · · 0 0 0:::

::::::

: : ::::

::::::

0 0 0 · · · χ1 χ2 00 0 0 · · · 0 χ1 χ2

:


As L•�−n� = HomR�L•�n�;R�, we see that H i�L•�−n�� = 0 for i 6= 0; 1,that H0�L•�−n�� ∼= R�−2n�, and that H1�L•�−n�� ∼= L ′�n��2�, where

L ′�n� = Homk

(R

�χ1; χ2�n; k

){− 2n+ 2}:

4.5.3. For n > 0 and λ ∈ �1k the module M�n; λ� of Subsection 4.3.3

produces the complex

M•�n; λ� = 0→ Rn�−2� → Rn→ 0

concentrated in complex degrees −1 and 0 with non-zero differential

χ1 + λχ2 0 0 · · · 0 0χ2 χ1 + λχ2 0 · · · 0 00 χ2 χ1 + λχ2 · · · 0 0:::

::::::

: : ::::

:::0 0 0 : : : χ1 + λχ2 00 0 0 · · · χ2 χ1 + λχ2

if λ 6= ∞;

χ2 0 0 · · · 0 0χ1 χ2 0 · · · 0 00 χ1 χ2 · · · 0 0:::

::::::

: : ::::

:::0 0 0 : : : χ2 00 0 0 · · · χ1 χ2

if λ =∞:

Clearly H i�M•�n; λ�� = 0 for i 6= 0. We claim that H0�M•�n; λ�� is isomor-phic to

M�n; λ� =

�χ1; χ2�n�χ1 + λχ2�n

�2n� if λ 6= ∞,

�χ1; χ2�n�χ2�n

�2n� if λ =∞.

This amounts to the exactness of M+�n; λ�, the complex obtained by aug-menting M•�n; λ� through the map εn;λ to M�n; λ� given on the canonicalbasis of Rn by

εn;λ�ei� ={ �−χ2�n−i+1�χ1 + λχ2�i−1 if λ 6= ∞,�−χ1�n−i+1χi−1

2 if λ =∞.


We have H1�M+�n; λ�� = 0 because εn;λ is surjective. The Euler charac-teristic of M+�n; λ� vanishes; hence so does that of its homology, provingH0�M+�n; λ�� = 0.

We return to the notation of Theorem 4.1.The following observation uses a variation of the argument of Buchsbaum

and Eisenbud for [13, (1.3)]; it is further generalized by Iyengar [23, (2.1)].

4.6

Let E be a minimal Q-free resolution of M with En = 0 for n > 2.For e = 1; 2 choose a homotopy σ�e�x E→ E between fe idE and 0E. We

have

∂σ�j�σ�i� = fjσ�i� − σ�j�∂σ�i� = fjσ�i� − σ�j�fi + σ�j�σ�i�∂ = fjσ�i� − fiσ�j�:When i = j the last expression vanishes. Otherwise, it equals −∂σ�i�σ�j�by symmetry. As ∂ is injective on E2 we get σ�j�σ�i��a� = −σ�i�σ�j��a� andσ�e�2�a� = 0 for a ∈ E0. The same relations hold trivially for a ∈ En withn 6= 0, so ξe�x� = σ�e��x� turns E into a DG module over K = R�ξ1; ξ2�by Example 1.6.

4.7

In view of the minimality of the Koszul resolution E constructed aboveand of Subsection 3.7, the complex of graded R-modules

C•�M;k� = 0→ R⊗k M2�−4� → R⊗k M1�−2� ∂→R⊗k M0 → 0

concentrated in degrees −2, −1, and 0, computes Ext∗R�M;k� by the for-mulas

ExtevenR �M;k� = H0�C•�M;k�� ⊕ H−2�C•�M;k��2�;

ExtoddR �M;k� = H−1�C•�M;k��1�:

When k is algebraically closed, Subsections 1.1 and 4.4 uniquely defineintegers: p; q�e�; r�e�; s�e�; t�0�; t�1�; t�2� ≥ 0;

integers: ae1; · · · ; aeq�e�; be1; · · · ; ber�e�; ce1; · · · ; ces�e� > 0;

pairs: �ce1; λe1�; · · · ; �ces�e�; λes�e�� with λej ∈ P1k;

for e = 0; 1, such that the graded T-module M = Ext∗Q�M;k� is isomorphicto

Kp ⊕q�0�⊕h=1

L�−a0h��−1� ⊕

r�0�⊕i=1

L�b0i � ⊕

s�0�⊕j=1

M�c0j ; λ

0j � ⊕ kt�0� ⊕ kt�2��−2�

⊕(q�1�⊕h=1

L�−a1h��−1� ⊕

r�1�⊕i=1

L�b1i � ⊕

s�1�⊕j=1

M�c1j ; λ

1j � ⊕ kt�1�

)�−1�:


The decomposition of M splits C•�M;k� into a direct sum of complexesof graded R-modules isomorphic to shifts of the complexes K•, L•�n�, andM•�n; λ� of Subsection 4.5. Thus, the graded R-module Exteven

R �M;k� isisomorphic to

kp ⊕q�0�⊕h=1

L ′�a0h� ⊕

r�0�⊕i=1

L�b0i � ⊕

s�0�⊕j=1

M�c0j ; λ

0j � ⊕ Rt�0� ⊕Rt�2��−2�

⊕q�1�⊕h=1

R�−2a1h − 2�

and the graded R-module ExtoddR �M;k� is isomorphic to(

q�1�⊕h=1

L ′�a1h� ⊕

r�1�⊕i=1

L�b1i � ⊕

s�1�⊕j=1

M�c1j ; λ

1j � ⊕ Rt�1�

)�−1�

⊕q�0�⊕h=1

R�−2a0h − 1� :

The multiplicities in the decompositions above satisfy some subtle re-lations. The first one comes from a well-known result of Auslander andBuchsbaum [3, (6.2)]:

4.8

As M is annihilated by a Q-regular element, we have∑qn=0�−1�n

βQn �M� = 0.

4.9

The numerical invariants appearing in Subsection 4.7 satisfy

q�0� + r�1� + t�1� = q�1� + r�0� + t�0� + t�2�:If ` denotes either side of the equality, then one of the following holds,

` = 0 = s�0� = s�1�;` = 0 < c�0� = c�1�;

` > a�1� − a�0� + b�1� − b�0� + c�1� − c�0� + q�0� + t�1� − t�0� > 0;

with a�e� =∑q�e�h=1 a

eh, b�e� =∑r�e�

j=1 bej , c�e� =

∑s�e�j=1 c

ej for e = 0; 1.

It is important to note that ` = 1 is excluded as a possibility, and it iscertainly interesting to know whether there are further restrictions.


To establish these relations, note that the decomposition of Ext∗Q�M;k�yields

βQ0 �M�=p+ a�0� + b�0� + r�0� + c�0� + t�0�;βQ1 �M�= 2p+ a�0� + q�0� + b�0� + c�0� + a�1� + b�1� + r�1�

+ c�1� + t�1�;βQ2 �M�=p+ a�1� + q�1� + b�1� + c�1� + t�2�:

The expressions for ` follow from the equality in Subsection 4.8.On the other hand, the decomposition of Ext∗R�M;k� yields for u � 0

equalities

βR2u�M�= `u− a�1� + b�0� + c�0� + t�0�;βR2u+1�M�= `u− a�0� + b�1� + c�1� + q�0� + t�1�:

When ` = 0 the Betti sequence of M is bounded, so by [5, (4.1)] it is even-tually constant. If that constant is zero, we get the first relation; otherwisefor u � 0 the formulas above give c�0� = β2u�M� and c�1� = βR2u+1�M�,and hence c�0� = c�1� > 0. When ` > 0 the Betti sequence of M is un-bounded, by [9, (7.8)] it is eventually strictly increasing; the formulas abovetranslate the inequalities βR2u+2�M� > βR2u+1�M� > βR2u�M� into the desiredinequalities for `.

4.10

For M as in Subsection 4.7, Proposition 1.2 yields

f-rankRM = f-rankT Ext∗Q�M;k�=p:

The free rank of M can also be read off the action of R on Ext¶3R �M;k�,

f-rankRM

= rankk Ker((χ1

χ2

)x Ext0

R�M;k�→ Ext2R�M;k�⊕ Ext2

R�M;k�)

−rankkCoker(�χ1 χ2�x Ext1

R�M;k� ⊕ Ext1R�M;k�

→ Ext3R�M;k�

):

Indeed, if a′ is the number of direct summands of Ext∗R�M;k� isomorphic toR�−3�, then by Subsection 4.7 the kernel has rank p+ a′ and the cokernelhas rank a′.


Proof of Theorem 4.1. Recall that M is a finite R-module withproj dimQM finite and g = depth R − depthR M. We want to provethat the R-module Ext∗R�M;k� is generated in degrees ≤ m, wherem = max�2βRg �M� + 1 ; 2βRg �M� + 2� + g.

A gth syzygy of M over R has finite projective dimension over Q anddepth equal to that of R; by Subsection 3.6 its cohomology over R is iso-morphic to a shift of Ext¾gR �M;k�. Switching from M to its syzygy, we mayassume that g = 0.

Choose a faithfully flat local homomorphism Q → Q′ such that ÎQ′ isthe maximal ideal of Q′ and k′ = Q′/ÎQ′ is algebraically closed, and setR′ = R ⊗Q Q′ = Q′/� f �, where f ′ is the image of f in Q′. For M ′ =M ⊗Q Q′ = M ⊗R R′ we then have a natural isomorphism of graded k′-vector spaces

Ext∗R�M;k� ⊗k k′ ∼= Ext∗R′ �M ′; k′�:If R′ is the ring of cohomology operators defined by the Q′-regular se-quence f ′, then the construction of the operators in Section 3 shows thatthis map is equivariant over an isomorphism R⊗k k′ ∼= R′. Switching fromQ to Q′ and changing notation once more, we may assume that the residuefield k is algebraically closed. The desired assertion is then obviously con-tained in the formulas in Subsection 4.7.

5. MINIMAL RESOLUTIONS

In this section �Q;Í; k� is a local ring, f = f1; : : : ; fc is a Q-regularsequence, R = Q/� f �, and M is a finite R-module of finite projective di-mension over Q. We explore the discrepancy between the R-free resolutionsG�E� constructed in Theorem 2.4 from Koszul resolutions E of M over Qand its minimal R-free resolution F.

It is clear that G�E� itself is minimal if and only if E is a minimal freeresolution of M over Q and each ξj acts trivially on Ext∗Q�M;k� (equiv-alently, on TorQ∗ �M;k�). However, the first condition is known to fail ingeneral; (cf. [4] or [7, Sect. 2]). When it does hold the second one usuallyfails. On the positive side, we note that M always has a Koszul resolutionof length equal to proj dimQM; cf. Subsection 2.1.

5.1

Lifting idM to quasi-isomorphisms αx F → G�E� and βx G�E� → Fone gets R-linear chain maps βχjαx F → F of degree −2. IdentifyingH∗HomR�F; k� with Ext∗R�M;k� we see that H HomR�βχjα; k� coincideswith the action of χj .


The homotopy class of each map βχjα is independent of the choices of α,β, or E; cf. Subsection 3.2. If there exist chain maps χ′j that are homotopicto βχjα and satisfy χ′iχ

′j = χ′jχ′i for 1 ≤ i; j ≤ c, then we say that F has

a proper structure of DG module over S . When c = 1 this condition isvacuous; in general, when it holds one gets on F a structure of DG moduleover S by letting χj act as χ′j .

5.2

A stronger requirement is that α can be chosen to satisfy χjα�F� ⊆ α�F�for 1 ≤ j ≤ c; we then say that F embeds as a DG submodule of G�E�; sinceF is minimal α defines an isomorphism F ∼= α�F� that induces a proper DGmodule structure on F. We make some general remarks on the existence ofsuch embeddings.

5.2.1. Let M = R and let K be a Koszul complex resolving R over Q.The inclusion R⊗ 1 ⊆ G�K� is then a quasi-isomorphism by Theorem 2.4. Itis obvious that this map splits over R and that its image is a DG submoduleover S .

5.2.2. Let M =M ′ ⊕M ′′. If E�i� is a Koszul resolution of M�i� and F�i�

is a minimal R-free resolution of M�i� that embeds as a DG submodule ofG�E�i��, then the minimal R-free resolution F′ ⊕ F′′ of M embeds as a DGsubmodule over S of G�E′� ⊕G�E′′� ∼= G�E′ ⊕ E′′�.

5.2.3. The minimal resolutions of the modules Ms described in Subsec-tion 4.2 admit no embedding as DG submodules of G�E�: this follows fromthe result of [9, (9.1)] that if such an an embedding exists, then the gradedR-module Ext∗R�M;k� is generated in degrees ≤ proj dimQM.

The next theorem is a partial converse to the result of [9] mentionedabove.

5.3. Theorem. Let M be a finite module over R = Q/�f1; f2�.If proj dimQM ≤ 2 then M has a minimal Koszul resolution E over Q.If furthermore Ext∗R�M;k� is generated over R in degree ≤ 2, then F is

isomorphic to a DG submodule of G�E� over S .

Eisenbud [18, p. 37] conjectured that the minimal resolution of each finiteR-module F can be embedded as a DG submodule of an R-free resolutionof M constructed from a system of higher homotopies on some Q-freeresolution of M; cf. Subsection 2.5. In [9, (9.3)] the conjecture is shownto fail, for the same reason as given in Subsection 5.2.3, but it is still notknown whether the conjecture can also fail for all high syzygies of a finiteR-module.

In codimension 1 the asymptotic conjecture holds for nth syzygies withn > depth R by Subsection 2.7. We prove it in codimension 2 with an


effective bound on n that depends on the module M. The modules Ms

show that no universal bound exists. In codimension ≥ 3 it is not knowneven whether F>n for n� 0 admits a DG module structure over S that isproper in the sense of Subsection 5.1.

5.4. Corollary. Let M be a finite module over R = Q/�f1; f2� withdepth R − depthR M = g. If proj dimQM < ∞ and M ′ is an nth syzygy ofM with

n > max{2βRg �M� − 1 ; 2βRg+1�M�

}+ g − 1 ;

then F>n is a DG submodule of G�E′� for a finite Koszul resolution E′ of M ′.

Proof. By Subsection 3.6, Ext¾nR �M;k��n� is isomorphic to the cohomol-ogy of the nth syzygy of M, so combine the preceding theorem with Sub-section 4.7.

Proof of Theorem 5.3. We note that the Koszul resolution E of M fromSubsection 4.6 is minimal, set E† = HomQ�E;R�, and use overlines forreduction modulo Î.

As the R-module Ext∗R�M;k� is generated in degrees ≤ 2, we see fromSubsection 4.7 that it has no direct summand isomorphic to L ′�a�; thismeans that Ext∗Q�M;k� = E† has no direct summand isomorphic to L�−a�with a > 0. By Subsection 5.2.2 and Subsection 5.2.1 we may further assumethat M has no direct summand isomorphic to R, and from Proposition 1.2we see that E† then has no direct summand of the form K.

Fix bases of the vector spaces L�b1i ��−1�2 for 1 ≤ i ≤ r�1�, and

M�c1j ; λ

1j ��−1�2 for 1 ≤ j ≤ s�1� and let ϒ2 be a lifting of that basis to

E†2. Similarly, choose bases in L�b0

i �1 for 1 ≤ i ≤ r�0� and in M�c0j ; λ

0j �1

for 1 ≤ j ≤ s�0� and lift them to a subset ϒ1 ⊂ E†1. It follows from

Subsection 4.5 that the subcomplex

D• = 0→ Rϒ2 → R ∂�ϒ2� ⊕Rϒ1 → R ∂�ϒ1� → 0

of graded R-modules of C• = C�E; k� ∼= C ⊗R k is exact, where C =C�ER;).

Recall that C\ = ⊕2p=0S ⊗R E†

p and let D be the DG submodule of Cgenerated over S by 1 ⊗ ϒe and 1 ⊗ ∂�ϒe� for e = 1; 2. Let D\ be thegraded R-module generated by χn1χn2 ⊗ v and ∂�χn1χn2 ⊗ v� with n1 ≥ 0,n2 ≥ 0, υ ∈ ϒ1 ∪ϒ2. These elements are linearly independent modulo ÍC,so they form a basis of D\ over R and D is a direct summand of C as acomplex of R-modules. The exactness of D• implies that D is split exact.

Set F = HomR�C/D;R� and note that HomR�C;R� is naturally isomor-phic as DG module over S to the R-free resolution G�E� of M from The-orem 2.4. Dualizing the exact sequence 0 → D → C→ C/D → 0 we get


an exact sequence

0→ F→ G�E� → HomR�D;R� → 0

of DG modules over S in which F splits off as a complex of R-modules.The complex HomR�D;R� is split exact along with D, so the sequence

shows that H∗�F� ∼= H∗�G�E�� ∼= M. The differential of the complex C/Dis trivial by the choice of D. As C/D ∼= �C/D�⊗R k, we see that ∂�F� ⊆ ÍF.Thus, F is a minimal free resolution of M and a DG submodule of G�E�over S .

Assume that one is able to construct finite sequences of syzygies overa residue ring R of codimension 2 over Q. When R is not a completeintersection, Iyengar [23, (3.4)] constructs a minimal R-free resolution of anR-module M, starting with a minimal Q-free resolution of its �depth R −depthR M + 2�nd syzygy. The preceding arguments yield a similar recipethat works when R is a complete intersection.

5.5

Let L be a finite module over R = Q/�f1; f2� with proj dimQ L <∞.Set g = depth R − depthR L and construct the beginning of a minimal

resolution of L up to the gth syzygy,

0→ L′ → Fg−1∂n−2→ Fg−2 → · · · → F1

∂1→ F0 → L→ 0:

The R-module L′ is free precisely when proj dimR L is finite. If that isnot the case, set n = max

{2βR0 �L′� ; 2βR1 �L′� + 1

}and extend the minimal

resolution to

0→M → Fn∂n→ Fn−1 → · · · → Fg+1

∂g+1→ Fg → L′ → 0:

Note that proj dimQM = 2, form a minimal Q-free resolution

0→ E2 → E1α→ E0 →M → 0;

and let u, v, w, be the ranks of E0, E1, E2. Choosing homotopies from f1 idE,f2 idE to 0 we get R-linear maps β�1�; β�2�x E0 → E1 and γ�1�; γ�2�x E1 → E2.

Let A, B�1�, B�2�, C�1�, C�2�, be, respectively, the matrices of the homo-morphisms α ⊗Q R, β�1� ⊗Q R, β�2� ⊗Q R, γ�1� ⊗Q R, γ�2� ⊗Q R in somebases of the free R-modules Eq ⊗Q R, subject to the only restriction thatthe last v − v′ rows of B = (B�1� ∣∣B�2�) form a k-basis of the row-space ofB. We describe the infinite tail F>0 of the minimal resolution of M in termsof the five matrices above.


Let C′ be a free graded R-module with basisχn11 χ

n22 υ

0i

with∣∣χn1

1 χn22 υ

0i

∣∣ = 2n1 + 2n2

χn11 χ

n22 υ

1h

with∣∣χn1

1 χn22 υ

1h

∣∣ = 2n1 + 2n2 + 1

∣∣∣∣∣∣∣∣∣∣1 ≤ i ≤ v′1 ≤ h ≤ u and n1; n2 ≥ 0

:The following set generates a homogeneous R-module direct summandD′ of C′,

u∑h=1

b�1�ih χ

n1+11 χ

n22 υ

1h

+u∑h=1

b�2�ih χ

n11 χ

n2+12 υ1

h

v′∑i=1

c�1�ji χ

n1+11 χ

n22 υ

0i

+v′∑i=1

c�2�ji χ

n11 χ

n2+12 υ0

i

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

v′ < i ≤ v

1 ≤ j ≤ wand n1; n2 ≥ 0

:

The formulas below define on F′ = C′/D′ an R-linear differential of degreeone,

∂�χn11 χ

n22 υ

0h� =

v′∑i=1

ahiχn11 χ

n22 υ

1i

∂�χn11 χ

n22 υ

1i � =

u∑h=1

b�1�ji χ

n1+11 χ

n22 υ

0h +

u∑h=1

b�2�ji χ

n11 χ

n2+12 υ0

h:

The R-dual of the complex �F′; ∂� is a minimal free resolution of M.

6. SPECTRAL SEQUENCE

Let f = �f1; : : : ; fc� be a Koszul-regular set in a commutative ring Q,let R = Q/� f �, and let S = R�χ1; : : : ; χc� be the ring of cohomology op-erators defined by f . This section describes our main tool for computingcohomology operators—a spectral sequence of S -modules that approxi-mates Ext∗R� ; �. The result is only stated in cohomology, as its homologicalversion is not used here.

6.1. Theorem. For all R-modules M, N there exists a spectral sequence

2Ep;q = S 2p+q ⊗R Ext−pQ �M;N� H⇒ Extp+qR �M;N�


with differentials r d of degree 1 for the cohomological degree p + q,and

r dp;qx rEp;q → rEp+r−1; q−r+2:

The spectral sequence is natural in both modules M and N.The rows rE∗; q are equal to 0 when q is odd, so

r d = 0 and rE = r+1E when r is odd:

The columns rEp; ∗ form complexes of graded S -modules

χj x rEp;q → rEp;q+2 and χjr d = r dχj:

The page 2E is the complex C•�M;N� of graded S -modules

· · · → S ⊗R Ext−pQ �M;N��2p�d→ S ⊗R Ext−p+1

Q �M;N��2p− 2� → · · ·

with differential d given by multiplication with∑cj=1 χj ⊗k ξj , and action of Λ

ξjx 2Ep;q → 2Ep+1; q−2 such that ξjd = −dξj and ξjχi = −χiξj:The abutment Ext∗R�M;N� has a filtration by graded S -submodules and⊕

p;q

∞Ep;q ∼= 0E�Ext∗R�M;N��

is an isomorphism of bigraded S -modules.

It is clear from the description of the second page of the spectral se-quence that outside of the sector p ≤ 0 and 2p + q ≥ 0 the sequencesatisfies rEp;q = 0 for all r ≥ 2. The resulting useful edge homomorphismsare described in Subsection 6.3.

As an illustration we give a transparent proof of the main result of [21](the converse holds as well and is proved in [9]). Note that Ext∗Q�M;N�(respectively, Ext∗R�M;N�) is killed by the annihilators of M and N in Q(respectively, R) so both Ext’s are naturally graded modules over the ring

R = Q/�annQM + annQ N� = R/�annRM + annR N�:The next corollary was initially proved by Gulliksen [21] (he uses a differ-

ent construction of the operators, but they coincide with those consideredhere, due to [11]).

6.2. Corollary. If the R-module Ext∗Q�M;N� is noetherian, thenExt∗R�M;N� is a noetherian graded module over R�χ1; : : : ; χc� = R⊗R S .


Proof. The hypotheses imply that 2E is a bigraded noetherian S -module, and 2Ep; ∗ = 0 for �p� � 0. It follows that rE is a bigraded noethe-rian S -module for each r ≥ 2, and rE = ∞E for r � 0. Thus, Ext∗R�M;N�is filtered by graded S -submodules and the associated bigraded module isnoetherian. We conclude that Ext∗R�M;N� is a noetherian graded moduleover S , and hence over R⊗R S .

Proof of Theorem 6.1. Let E be a Koszul resolution of M. For the DGmodule C�E;N� with C�E;N�\ = ⊕∞u=0S ⊗R HomQ�Eu;N��−2u� fromSubsection 3.2 set

F�p� = ⊕u+v¾−p

u¾0

S v ⊗R HomQ�Eu;N��−2u�:

This is a descending filtration of C�E;N�, graded by cohomologi-cal degree, by means of DG submodules over S . It defines a spec-tral sequence with differentials r dp;qx rEp;q → rEp+r−1; q−r+2. SinceC�E;N� ∼= HomR�G�E�;N�, where G�E� is the R-free resolution of Mgiven by Theorem 2.4, the spectral sequence converges to Ext∗R�M;N� forthe cohomological degree p+ q. Its 0th page has the form

0Ep;q = (F�p�/F�p− 1�)p+q = S 2p+q ⊗P HomQ�E−p;N�with differential 0 dp;q = S 2p+q ⊗R HomQ�∂E

−p;N�. As S 2p+q is a free R-module, we have 2Ep;q = 1Ep;q = S 2p+q ⊗R Ext−pQ �M;N� and 2 dp;q =∑cj=1 χj ⊗ ξj .When q is odd 0Ep;q = 0. It follows that if r is odd then r d = 0, so

r+1E = rE. Furthermore, the construction shows that the algebra S ⊗R Λoperates on the spectral sequence by χj · rEp;q ⊆ rEp;q+2 and ξj · rEp;q ⊆rEp+1; q−2.

6.3

The preceding proof yields useful information on the edge homomor-phisms of the spectral sequence of Theorem 6.1, determined by its vanish-ing lines

rEp;q = 0 when 2p+ q < 0 or p > 0:

6.3.1. The first vanishing line defines diagonal edge homomorphisms

ExtuR�M;N�� ∞E−u; 2u ↪→ · · · ↪→ 3E−u; 2u ↪→ 2E−u; 2u = ExtuQ�M;N� ytheir composition is the component ρuMN of the change-of-rings morphismfrom Subsection 3.5.1 and 4E−u; 2u = 3E−u; 2u = annExtuQ�M;N��ξ1; : : : ; ξc�.


6.3.2. The second vanishing line defines vertical edge homomorphisms

S 2v ⊗R Ext0Q�M;N� = 2E0; 2v� 3E0; 2v� · · ·� ∞E0; 2v ↪→ Ext2v

R �M;N� ytheir composition is the component ~2v

MN of the morphism from Subsec-tion 3.5.2.

6.3.3. If ExtnQ�M;N� = 0 for n > g, then rEp;q = 0 for p < −g; itfollows that rEp;q = ∞Ep;q for r > g + 1 and so there exist vertical edgehomomorphisms

ExtvR�M;N�� ∞E−g; v+g ↪→ · · · ↪→ 3E−g; v+g ↪→ 2E−g; v+g

= S v ⊗R ExtgQ�M;N�ytheir composition is a component of a morphism of graded S -modules

Ext¾gR �M;N� → S ⊗R ExtgQ�M;N��−g�:

6.4

The spectral sequence allows for some extensions of earlier results.

6.4.1. If f = f1 and ExtpQ�M;N� = 0 for p > 1, then the line 2E∗; 2v ofthe spectral sequence of Theorem 6.1 becomes the complex

0→ Rχq1 ⊗R Ext1

Q�M;N� Rχq+11 ⊗R Ext0

Q�M;N� → 0:χ1⊗ξ1

We see that 3Ep;q = ∞Ep;q and ∞Ep;q = 0 for �p; q� 6= �1; 2i− 1�; �0; 2i�;hence

ExtnR�M;N�

∼=

HomR�M;N� = HomQ�M;N� for n = 0,Ker

(ξ1x Ext1

Q�M;N� → HomQ�M;N�)

for n = 2i > 0,Coker

(ξ1x Ext1

Q�M;N� → HomQ�M;N�)

for n = 2i+ 1 > 0,

where the map χ1x ExtnR�M;N� → Extn+2R �M;N� is the obvious epimor-

phism in degree 0 and the obvious isomorphism in positive degrees.In view of Subsection 3.6, the preceding computation yields an ef-

fective version of Gulliksen’s finiteness result, Corollary 6.2, above: ifExtpQ�M;N� = 0 for p > g + 1 then the R�χ1�-module Ext∗R�M;N� isgenerated in degrees ≤ g, and χ1 is regular on Ext>gR �M;N�. Whenproj dimQM is finite this follows also from Subsection 2.7.

6.4.2. If ExtpQ�M;N� = 0 for p > 2, then there is an exact sequence

0→ H0C•�M;N� → ExtevenR �M;N� → H−2C•�M;N��2� → 0

of graded S -modules, and an isomorphism of graded S -modules

ExtoddR �M;N� ∼= H−1C•�M;N��1�:

When Q is local and N is its residue field k, a slightly more precisestatement provided the starting point of the computation in Subsection 4.7.


7. FINITE CI-DIMENSION

Let R be a local ring with maximal ideal Í and residue field k.Recall from [9] that a quasi-deformation of R (of codimension c) is a pair

of local homomorphisms R → R′ ← Q, the first being faithfully flat andthe second surjective with kernel generated by a Q-regular sequence f (oflength c). An R-module M has finite CI-dimension, denoted CI-dimRM <∞, if the Q-module M ′ =M ⊗R R′ has finite projective dimension for somequasi-deformation of R.

7.1

If F is a minimal free resolution of M over R, then F ⊗R R′ is onefor M ′ over R′, so βnR�M� = βnR′ �M ′� for all n, hence cxR M = cxR′M ′.When proj dimQM

′ is finite, M′ = Ext∗R′ �M ′; k′� is a finite module overR′ = k′�χ1; : : : ; χc� by Corollary 6.2. As R′ lives in even degrees we get adirect sum of finite graded R′-modules

Ext∗R′ �M ′; k′� = ExtevenR′ �M;k� ⊕ Extodd

R′ �M ′; k′�:

By the Hilbert–Serre theorem, for n� 0 the sequence of even (respectivelythe sequence of odd) Betti numbers is given by some polynomial in n.

The following basic result (cf. [5, (3.6)] or [9, (5.10)]), provides eachmodule of finite CI-dimension with a “best” quasi-deformation.

7.2

When CI-dimRM < ∞ there exists a quasi-deformation R → R′ ← Qof codimension d = cxR M such that proj dimQ�M ⊕R R′� <∞, edimQ =edimR′, and the residue field of Q is algebraically closed.

Next we give a new proof of [5, (4.1)]. As the original one, the argumentpresented here starts from the construction above; we believe that the routeit takes from there is more enlightening (if less elementary). By convention,0! = 1.

7.3. Theorem. Let M be a finite R-module of finite CI-dimension andinfinite projective dimension, and set d = cxR M. There exist a positive integerβ-degR�M� and polynomials peven�t� ; podd�t� ∈ ��t� of degree ≤ d− 2 suchthat

βRn �M� =β-degR�M�2d−1�d − 1�! n

d−1 +{peven�n� for even n� 0,podd�n� for odd n� 0.


We call β-degR�M� the Betti degree of the R-module M.

7.4. Example. In case R = Q/� f � with proj dimQM = q < ∞, letβQ�M� = ∑q

n=0 βQn �M� be the total Betti number of M over Q. The re-

sult in [7, (3.1.3)] yields a coefficientwise inequality of formal power series

∞∑n=0

βRn �M� tn´q∑n=0

βQn �M� tn/�1− t2�c

with equality if f ⊂ Î annQ�M�. For 2u ≥ q the coefficient of t2u on theright-hand side has the form �b/�c − 1�!�uc−1 + p�u� with b = ∑i β

Q2i�M�

and p a polynomial of degree < c − 1. By Subsection 4.8 we have b =βQ�M�/2; hence

β-degRM = 2c−1�c − 1�! limu→∞

βR2u�M��2u�c−1 ≤ lim

u→∞βQ�M�uc−1 + 2p�u�

2uc−1

= βQ�M�

2

and equality holds if the regular sequence f is contained in Î annQ�M�.Proof of Theorem 7.3. In view of Subsection 7.2 we may assume that

R = Q/� f � for a Q-regular sequence f = f1; : : : ; fd with d = cxR M andproj dimQM < ∞. It is clear from Subsection 7.1 that d is the greater ofthe degrees of the polynomials qeven and qodd that give the even and oddBetti numbers for n � 0. We have to prove that these polynomials havethe same degree and equal leading coefficients.

Let R = k�χ1; : : : ; χc� be the ring of cohomology operators acting onExt∗R�M;k�. For reven = rankR Exteven

R �M;k� and rodd = rankR ExtoddR �M;k�

we have

reven = �d − 1�! limu→∞

βR2u�M�ud−1 = 2d−1�d − 1�! lim

u→∞βR2u�M��2u�d−1 ;

rodd = �d − 1�! limu→∞

βR2u+1�M�ud−1 = 2d−1�d − 1�! lim

u→∞βR2u+1�M��2u+ 1�d−1 ;

with the equalities on the left given by standard multiplicity theory. On theother hand,

reven − rodd = rankR

( ⊕p even

∞Ep;∗)− rankR

( ⊕p odd

∞Ep; ∗)

(a)

=∑p

�−1�prankR∞Ep; ∗ (b)


=∑p

�−1�prankR1Ep; ∗ (c)

=∑p

�−1�pβQp �M� (d)

= 0 (e)

where (a) is due to the convergence of the spectral sequence of R-modulesof Theorem 6.1, (b) is due to the additivity of rank, (c) is due to the per-manence of Euler characteristics, (d) is due to the complex C•�M;k� ofTheorem 6.1, and (e) is given by Subsection 4.8. Altogether,

2d−1�d − 1�! limn→∞

βRn �M�nd−1 = reven = rodd:

Thus, qeven and qodd have equal leading term of degree d as desired.

7.5. Conjecture. If M is a finite R-module of infinite projective dimen-sion and finite CI-dimension with cxR M = d, then β-degR�M� ≥ 2d−1, or,equivalently,

limn→∞

βRn �M�nd−1 ≥

1�d − 1�! :

7.5.1. The conjecture holds for modules of complexity ≤ 2.Indeed, when cxR M = 1 the Betti numbers of M are constant for n� 0,

so β-degR�M� = βRn �M� ≥ 1. When cxR M = 2 and in addition R =Q/�f1; f2� with proj dimQM finite and k algebraically closed, then Sub-section 4.9 yields limn β

Rn �M�/n = `/2 and ` ≥ 2; Subsection 7.2 reduces

the general case of complexity 2 to the special one.

7.5.2. Let M be a Q-module of finite projective dimension that is an-nihilated by a Q-regular sequence of length c. The squares of the ele-ments in the given sequence form a Q-regular sequence f in Î annR�M�,so β-degRM = βQ�M�/2 by Example 7.4, and the conjecture predicts aninequality βQ�M� ≥ 2c. The last inequality is proved in [8] when c ≤ 5, orwhen M is a graded module of odd multiplicity over standard graded poly-nomial rings. It would follow in general from the lower bounds βQn �M� ≥(cn

)conjectured by Buchsbaum and Eisenbud [13, (1.2)].

7.5.3. If g = g1; : : : ; gc is a Q-regular sequence with f ⊂ Î�g�, thenM = Q/�g� has cxMR = c and β-degRM = 2c−1, so the bound in Conjec-ture 7.5 is sharp.

The critical degree cr degRM of a finite R-module M is defined in [9,Sect. 7] to be the infimum of those s ∈ � for which the minimal resolutionF of M admits a chain map µx F→ F of degree q < 0 with µ�Fn+q� = Fn


for n > s. Clearly, cr degRM ≤ proj dimRM with equality if the latter isfinite.

The critical degree is significant because it may be finite even when theprojective dimension is not, and then it yields data on the growth of Bettinumbers: when CI-dimRM <∞ and depth R− depthR M = g it is provedin [9, (7.3)] that

• if cxR M ≤ 1 then cr degRM = s ≤ g and βRn �M� = βRn+1�M� forn > s,

• if cxR M ≥ 2 then cr degRM = s < ∞ and βRn �M� < βRn+1�M� forn > s.

We give an effective bound on the critical degree for modules of finiteCI-dimension and complexity 2. No bound is known in complexity ≥ 3.

7.6. Theorem. If M is a finite R-module with CI-dimRM < ∞ andcxR M = 2, then cr degRM ≤ max�2βRg �M� − 1; 2βRg+1�M�� + g − 1 forg = depth R− depthR M.

Proof. Composition products turn Ext∗R�k;k� into a graded algebraand Ext∗R�M;k� into a graded left module over it. Let P be the k-subalgebra of Ext∗R�k;k� generated by the central and primitive elementsof Ext2

R�k;k�. By [9, (7.2)] we have to show that depthP Ext¾nR �M;k� > 0for n > max�2βRg �M� − 2; 2βRg+1�M� − 1� + g.

Using Subsection 7.2, choose a quasi-deformation R→ R′ ← Q such thatR′ = Q/�f1; f2� has an algebraically closed residue field k′ and M ′ =M ⊗RR′ satisfies proj dimQM

′ <∞. There results an equivariant isomorphism

Ext∗R�M;k� ⊗k k′ ∼= Ext∗R′ �M ′; k′�of graded left modules over the isomorphism of graded k′-algebras

Ext∗R�k;k� ⊗k k′ ∼= Ext∗R′ �k′; k′�:The last map sends P ⊗k k′ isomorphically onto the k′-subalgebra P ′

generated by the central and primitive elements of Ext2R′ �k′; k′�. As

Ext¾nR �M;k� is finite over P by [11, (5.3)], we have depthP ′ Ext¾nR′ �M ′; k′� =depthP Ext¾nR �M;k�, so adjusting notation we may assume that R =Q/�f1; f2� and k is algebraically closed.

Let R = k�χ1; χ2� be the ring of cohomology operators acting onExt∗R�M;k�. By [11, (3.3), (4.2)] there is a homomorphism of gradedk-algebras R → P that is compatible with both actions on Ext∗R�M;k�.The R-module Ext¾nR �M;k� is finite by Corollary 6.2, so it has the samedepth over P and over R. The structure of the R-module Ext∗R�M;k� de-termined in Subsection 4.7 shows that it contains no non-zero submoduleof finite length when n > max�2βRg �M� − 2; 2βRg+1�M� − 1� + g.


7.7

The preceding argument shows that when M is a finite R-module offinite CI-dimension the P-module Ext∗R�M;k� is generated in degree ≤cr degRM + 3. By [9, (4.9)] it is generated in the same degrees also asa module over the R-subalgebra of Ext∗R�M;M� spanned by the centralelements in Ext2

R�M;M�.

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