JOURNAL OF ALGEBRA 190, 195᎐213Ž. 1997 ARTICLE NO. JA966901
Free Resolutions and Change of Rings
Srikanth Iyengar*
Department of Mathematics, Purdue Uni¨ersity, West Lafayette, Indiana 47907
Communicated by Da¨id Buchsbaum View metadata, citation and similar papers at core.ac.uk brought to you by CORE Received February 25, 1996 provided by Elsevier - Publisher Connector
Projective resolutions of modules over a ring R are constructed starting from appropriate projective resolutions over a ring Q mapping to R. It is shown that such resolutions may be chosen to be minimal in codimension F 2, but not in codimension 3. This is used to obtain minimal resolutions for essentially all modules over localŽ. or graded rings R with codimension F 2. Explicit resolutions are given for cyclic modules over multigraded rings, and necessary and sufficient conditions are obtained for their minimality. ᮊ 1997 Academic Press
INTRODUCTION
We construct a projective resolution FXŽ.,Y of a module M over a ring R, starting from appropriate projective resolutions, X of R and Y of M, over a ring Q mapping to R. The interest in such a procedure comes from the fact that homological properties of modules over Q are often better understood. A typical case is when R is a quotient of a polynomial ring Q: resolutions over Q are then finite, but those over R are usually infinite. Thus, one can build, in a finite number of steps, suitable Q-resolutions of R and M, and then use our construction to get a resolution over R. When Q is a field, the classical bar construction of R and M over Q yields a projective resolution of M over R. In Section 1 we generalize this by using differential gradedŽ. henceforth abbreviated DG bar construc- tions to obtain a resolution FXŽ.,Y of M over R. As an application, we give a short construction of a spectral sequence of Avramov. Our construction can be used to obtain a free R-resolution of M only if both Q-resolutions admit appropriate DG structures, so we consider the
* E-mail: [email protected].
195
0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. 196 SRIKANTH IYENGAR following question: If X is a Q-free resolution of R with a structure of a DG algebra, then does a gi¨en Q-free resolution Y of M admit a structure of a DG module o¨er X ? The work of Avramov, Buchsbaum, Eisenbud, Kustin, Miller, Srinivasan, and others provides answers in many situations. When Ž. Xis a Koszul complex, the answer is positive for M s QrI with pdQ QrI Ž. F3, or pdQ QrI s 4 and QrI Gorenstein, and may be negative when Ž. Ž. pdQQQrI s 4, or pd QrI s 5 with QrI Gorenstein. In view of these ‘‘positive’’ and ‘‘negative’’ results, the only unresolved case is that of a module M having a resolution of length at most 3. In Section 2, we prove that if Y has length F 2, then Y has a DG module structure over any DG algebra X resolving R. On the other hand, using Avramov’s obstructions, we show that this result cannot be extended to resolutions of length 3, even when X is a Koszul complex on a single nonzero di¨isor. When dealing with localŽ. or graded rings, the objective is to build minimal resolutions. In general, FXŽ.,Y is nonminimal over R, even if both resolutions X and Y are minimal over Q. Nevertheless, in codimen- sion F 2 the construction can be used to write down the minimal resolu- tion of an a priori specified syzygy module of each module M over R.
When pdQ R s 1, we recover the periodic of period 2 resolutions con- structed by Shamash and Eisenbud. Moreover, when edim R y depth R F 2 and R is presented as a quotient of a regular local ring, the entire minimal free resolution of an R-module M can be obtained in a finite number of steps. This is the content of Section 3. For multigraded quotients of polynomial rings, there is an explicit, though rarely minimal, DG algebra resolution: the Taylor resolution. In Section 4, we use it to construct resolutions of cyclic modules over multigraded rings. Thus, we recover a result of Charalambous, who constructed by a different procedure, a free resolution of the residue field over a multigraded ring. This result provided the starting point for our investigations.
1. RESOLUTIONS
Ž.1.1 Differential graded algebras. Let Q be a commutative ring. For a complex of Q-modules W, we write Wh for the underlying graded Q-mod- ule, and < Ž.XЈ,ѨЈof DG algebras is a homomorphism of the underlying graded algebras with Ѩ Ј s Ѩ. For a DG algebra X, we denote the complex of Q-modules CokerŽ.Q ª X by X. A DG module Y over a DG algebra X is a complex Ž.Y, Ѩ with Yha Ž.1.2 THEOREM. Let R be a Q-algebra and M an R-module. Let X be a Q-projecti¨e DG algebra resolution of R with X0 a projecti¨eQ-module. Let Y be a Q-projecti¨e resolution of M such that Y is a DG module o¨er X. There is a projecti¨e resolutionŽŽ F X, Y ., Ѩ .of M as an R-module with FXniŽ.,Ys RmXmиии m Xim Yj, [ 1p pqi1qиии qi pqjsn and Ѩ s Ѩ Ј q Ѩ Љ, where p eiy1 Ѩ ЈŽ.x1m иии m x pm y syÝŽ. y1 x1mиии m ѨXiŽ.x m иии m x pm y is1 ep qyŽ.1 x1mиии m x pYm Ѩ Ž.y , py1 e ѨЉиии i иии иии Ž.x1mm rpm y syÝŽ.1x1mm xxiiq1m m x pm y is1 1e qpy1 иии qyŽ.1 x1m m x py1m xyp with ei s i q < Ž.1.3 Bar constructions. Let X be a DG algebra, and let W and Y be DG X-modules. A complex of Q-modules  QŽ.W, X, Y is defined as follows. The underlying graded Q-module has Q nkŽ.W, X, Y s W m Xim иии m Xim Yj. [1p pqkqi1qиии qi pqjsn It is generated by elements w m x1 m иии m x p m y s wxw 1 <<иии xyp x. The Q differential on  Ž.W, X, Y is given by Ѩ s Ѩ Ј q Ѩ Љ, where ѨЈŽ.wx1<<иии xyp p eiy1 sѨWŽ.wx1<<иии xyp yyÝŽ.1wx1<<иии ѨXiŽ.x <<иии xy p is1 ep qyŽ.1 wx1<<иии x pYѨ Ž.y ѨЉŽ.wx1<<иии xyp py1 ee 0<<иии i<<иии <<иии syŽ.Ž.1 wx12 x xypqyÝŽ.1wx1 xxiiq1 xyp is1 1e qpy1 <<иии qyŽ.1 x1 xxypy1Ž.p , i with eijs i q < The complex of Q-modules  QŽ.X, X, Y is a DG module over X, with operation on the left-hand factor. It is called the bar construction of Y over the DG Q-algebra X. Ž.1.4 THEOREM. Let X s ŽX, Ѩ .be a Q-projecti¨e DG algebra resol¨ing Ž. R and such that X0 is a projecti¨eQ-module. Let Y s Y, Ѩ be a Q-projec- ti¨e resolution of M such that Y is a DG X-module. Q The complex  Ž.R, X, Y is a projecti¨e resolution of M as an R module. If X and Y are Q-free, then  QŽ.R, X, YisanR-free resolution of M. QŽ. Ž.Ž Proof. Define ⑀:  X, X, Y ª Y by ⑀ xywx sxиyand ⑀ xxw1 <иии . < xyp x s0 for p ) 0, and note that ⑀ is a morphism of complexes. Q Let : Y ª M be the augmentation map, and let :  Ž.X, X, Y ª Q Ž.R,X,Ybe the map induced by the augmentation X ª R. Set s ⑀. FREE RESOLUTIONS AND RINGS 199 In the commutative diagram QQŽ. Ž.⑀ 6 R,X,Y6 X,X,YY 6 6 RmX6 MMM is a quasi-isomorphism. Thus, to prove that R mX is a quasi-isomor- phism, it is enough to show that ⑀ and are quasi-isomorphisms. The Q-linear homomorphisms Q Q Q : Y ª  Ž.X, X, Y , :  Ž.Ž.X, X, Y ª  X, X, Y , Ž.y s1wxy, Ž.xx1<<иии xyps1xx <<<1иии xypfor p G 0 Ž. satisfy ⑀ s 1YB and Ѩ q Ѩ s 1 y ⑀. It follows that H ⑀ is an isomor- phism. QŽ. Filter the complex C s  X, X, Y by the number of bars; thus FCp is the submodule generated by elements xxw 1 <<иии xyi xwith i F p. Let Ä4  QŽ. FDppG0be the corresponding filtration on D s R, X, Y . In the spectral sequences associated to the filtered complexes C and D, we have EC00Ž. X Land EDŽ. R L, where L Ž.X mpY . p, ) smQp p,) smQp ps m Q Note that : C ª D is filtration preserving, and hence yields a homo- morphism HX L E1 Ž.C E1 Ž.D HR L. qppž/s ,qpª ,qqps ž/ mmQQ As Xhhand Y are Q-projective, Lp is a bounded below complex of projective modules. Thus the quasi-isomorphism X ª R induces a quasi- isomorphism X L R L .So EC1 Ž. ED1 Ž., and hence mQpª m Qp p,) (p,) HŽ.:HC Ž.(HD Ž .is an isomorphism. To complete the proof, note that  QŽ.R, X, Y is a complex of projective Ž.respectively, free R-modules, and hence a projective Ž. respectively, free resolution of M. Ž.1.5 Proof of Theorem Ž.1.2 . The formulas in Ž. 1.3 for the differentials of the bar construction show that the complex of R-modules underlying QŽ.R,X,Yis exactly the complex inŽ. 1.2 . As an application ofŽ. 1.5 we obtain a spectral sequence and a sufficient condition for its collapse, given inw 2,Ž.Ž. 3.1 , 4.1x . The sequence below starts on the first page, rather than the second: this is useful in characteriz- ing Golod modules; cf.Ž. 1.7 200 SRIKANTH IYENGAR Ž.1.6 THEOREM. If Q ª R is a finite homomorphism of local rings, inducing the identity on the common residue field k, and M is a finite R-module, then there exists a first quadrant spectral sequence con¨erging to Tor RŽ.M, k . The first page of the spectral sequence is the standard reduced bar construction, with p E1 Tor QQŽ.R, k m Tor ŽM, k ., p,qsž/ mk q py1 dx1<<иии xy 1ix<<иии xx <<иии xy p,)Ž.1psyÝŽ.1iiq1 p is1 p <<иии qyŽ.1 x1 xxypy1Ž.p , QQ where Tor Ž.R, k s Coker Žk ª Tor Ž..R, k . The second page of the spec- tral sequence is 2 Tor QŽ R , k. Q Ep,qps Tor Ž.k, Tor Ž.M, k q. If the minimal Q-free resolution of R has a structure of a DG algebra, and the minimal Q-free resolution of M has a DG module structure o¨er it, then 2 ϱ the spectral sequence collapses with E s E . Proof. Choose a DG Q-algebra X and a DG X-module Y satisfying the hypothesis ofŽ. 1.4 . The complex  QŽ.R, X, Y is then an R-projective R resolution of M, and Tor Ž.Ž.M, k s HC, where Q k C s k  Ž.R, X, Y s  Ž.k, Ž.Ž.X m k , Y m k . mR Consider the spectral sequence associated to the filtration of C defined by Q the number of bars. Noting that HXŽ.mksTor Ž.R, k and HY Žmk . Q 1 sTor Ž.M, k , the Kunneth¨ formula gives the E page. As k is a field, the 2 Ž 1. Ž. expression for Ep, qpqs w HExwresults from 20, Chap. X, 8.2x . To complete the proof: if X and Y are minimal, then X m k and Y m k 2 ϱ have zero differential, and hence E s E . Ž.1.7 Golod modules. Let : Q ª R be as in Ž. 1.6 . For a finitely generated R-module M, the Poincare´ series of M is the formal power RŽ. Ý i  series PtMis G0iit, where the ith Betti number is the rank of the ith free module in a minimal free resolution of M. Let X be a DG algebra and Y a DG module over X satisfying the hypothesis of TheoremŽ. 1.4 . The convergence of the spectral sequence in Ž.1.6 gives ϱ 1 0 ÝÝÝdim kpE ,qkpF dim E ,qkpF dim E ,q. pqqsnpqqsnpqqsn FREE RESOLUTIONS AND RINGS 201 These numerical inequalities yield coefficientwise inequalities of formal power series P Q Ž.t Ý rank Yti R MiG0Ž.Qi PtMŽ.UU . 1 tPQŽ. t 1 Ý i yŽ.Ry 1ytŽ.iG0Ž.rank QiXty1 A module M for which equality holds on the left is called -Golod by Levinwx 19 . This is equivalent to the condition that the spectral sequence in Ž.1.6 collapses on the first page. If the R-module k is -Golod, then is said to be a Golod homomorphism. Note that equality holds on the right if and only if X and Y are minimal over Q. We are now in a position to prove the following Ž.1.8 PROPOSITION. Let X and Y be minimal Q-free resolutions of R and Ž. M,respecti¨ely, and set Xqs Ker X ª k . The following conditions are equi¨alent: Ž.i M has a minimal resolution of the form F Ž X, Y .. Ž. и ii X has a DG algebra structure with XqqX ; ᒊ X q, and Y has a и structure of a DG X-module with Xq Y ; ᒊY. Ž.iii X has a structure of a DG algebra; Y has a structure of a DG X-module, and M is -Golod. Proof. The equivalence ofŽ. i and Ž ii . follows from the expression for the differential on FXŽ.,Y, given in Ž. 1.2 . Assume that X and Y have the appropriate DG structures. The resolu- tion FXŽ.,Y is minimal over R if and only if both inequalities of formal power series above become equalities. ByŽ. 1.7 , as X and Y are minimal, this is equivalent to the condition that M is -Golod. Thus,Ž. i m Žiii . . 2. EXISTENCE OF MULTIPLICATIVE STRUCTURES Let Q be a noetherian local ring and I an ideal in Q. When I is an ideal generated by a regular sequence, the Koszul complex resolving QrI has a DG algebra structure. This raises the question, first asked by Buchsbaum and Eisenbudwx 8 , as to whether the minimal resolution of QrI admits a DG algebra structure in general. They give a positive answer when pdQ QrI F 3. When pdQ QrI s 4 and QrI is Gorenstein, such an answer was obtained by Kustin and Millerwx 17, 16 . A related question, also raised inwx 8 , is the following: Let J be an ideal generated by a regular sequence contained in I. Suppose that X is the Koszul complex resolving QrJ. Does the minimal resolution Y of QrIo¨er Qha¨e a structure of a DG X-module? This is a weaker question: Indeed, if 202 SRIKANTH IYENGAR Y has a DG algebra structure, then by the universal property of Koszul complexes, it has the structure of a DG X-module. Thus, in view of the discussion above, Y has a DG module structure over X when pdQ QrI F 3 or pdQ QrI s 4 and QrI is Gorenstein. On the other hand, Srinivasanwx 25 gives an example of an ideal I such that the minimal resolution Y does not admit the structure of a DG algebra, but for every ideal J : I generated by a Q-regular sequence, Y has the DG module structure over the Koszul complex resolving QrJ. In the light of these results, we begin with the following proposition, the proof of which uses an idea fromw 8,Ž. 1.3x : Ž.2.1 PROPOSITION. Let : Q ª R be a surjecti¨e homorphism of rings, and let X be a Q-free DG algebra resolution of R, with X0 s Q. If Y is a Q-free resolution of M such that Yi s 0 for i ) 2, then Y has the structure of a DG module o¨er X. Proof. Let : R m M ª M be the structure map given by Ž.r m m s rm. By the classical comparison theorem for resolutions, there is a mor- phism of complexes : X m Y ª Y such that the following diagram, where the vertical maps are the canonical augmentations, commutes: 6 X m YY 6 6 , 6 RmMM Furthermore, we can choose to extend the canonical isomorphism ( Y s X0 m Y ª Y. Define the action of X on Y by x и y s Ž.x m y for x g X and y g Y. The only thing that needs to be proved is that the action is associative, that Ž.Ž. is, x12и x и y s xx 12иy for x 12, x g X and y g Y. By construction X0 mYªYis the identity map on Y, and hence, by degree considera- tions, we need only deal with x12, x g X 1and y g Y0. In this case, x12и Ž.Ž.x и y and xx 12иyare in Y2. As the differential of Y Ž Ž .. ŽŽ . . is injective on Y21, it is enough to prove that Ѩ x и x2и y s Ѩ xx12иy. A direct computation shows that Ѩ Ž.x12и Ž.x и y s Ѩ Ž.Ž.Ž.Ž.x1 x 2m y y Ѩ x2 x 1m y s Ѩ Ž. Ž.xx12иy, thus proving associativity. In attempting to extendŽ. 2.1 , one has to contend with the following result proved by Avramovwx 2 . Let Q be an arbitrary local ring and m and n be integers subject to the conditions depth Q G n G m G 2. If n G 4, then there exists a perfect ideal I FREE RESOLUTIONS AND RINGS 203 in Q of grade n, and an ideal J in I generated by a Q-regular sequence of length m, for which the minimal resolution of M s QrI does not admit a DG module structure o¨er the Koszul complex resol¨ing R s QrJ. If n G 6, then the ideal I can be chosen to be Gorenstein. In addition, a recent paper by Srinivasanwx 26 contains an example of a grade 5 Gorenstein ideal I and an ideal J generated by a regular sequence in I, such that the minimal resolution of QrI does not admit a structure of a DG module over the Koszul complex resolving QrJ. The following theorem shows thatŽ. 2.1 cannot be extended to length 3 resolutions: Ž.2.2 THEOREM. Let Q be an arbitrary local ring and let n be an integer such that depth Q G n G 3. There exists a perfect Q-module L of grade n and a nonzero di¨isor f g Q in the annihilator of L, for which the minimal resolution of L does not admit a DG module structure o¨er any DG algebra resol¨ing R s Q r ()f . The proof of the theorem is given inŽ. 2.6 . It uses some facts about obstructions to DG structure on minimal resolutions, which we now recall. Ž.2.3 Obstructions. Let : Q ª R be a finite local homomorphism, inducing the identity map on their common residue field k, and let M be Q Ž. an R module. Denote by Torq R, k the kernel of the composition QŽ. QŽ. Tor R, k ª Tor0 R, k s R m k ª k. Inwx 2 Avramov defines a graded k-module oM Ž., called the obstruction to the existence of multiplicative structure, as the kernel of the canonical map of graded k-spaces Tor Q Ž.M, k Tor RŽ.M, k . QQи ª TorqŽ.R, k Tor ŽM, k . Supposing that the minimal Q-free resolution X of R has the structure of a DG algebra, he proves the following result: If oM Ž./0, then the minimal Q-free resolution of M does not admit the structure of a DG module over X; cf.w 2,Ž. 1.2x . Ž22. When Q s kxwx1234,x,x,x and R s Qr x14, x , Avramov shows that Ž. Ž22. oM/0 for M s Qr x11223344, xx,xx,xx,x . Using this example and the functorial properties of oŽ.᎐ established inwx 2 , he constructs a grade 4 perfect module with a nontrivial obstruction. We construct a grade 3 example by a similar procedure. 204 SRIKANTH IYENGAR Ž.2.4 Construction. Over the polynomial ring QЈ s kawx,b,c, the com- plex 0 c2 ac 00 0 2 b0a0c2 yb c 0ybc a 0 ycya 0 c 00 0 0 2 6 6 ž/6 ž/c a b 0 ca0c 0 y 35y y 3 G:0ªQЈ QЈ QЈ QЈ 5 is exact and minimal. Consider the multigraded module NЈ s CokerŽQЈ 3. ª QЈ , which is generated by elements e123, e , e with multidegrees 55e123sŽ.0,y1,1 , 55e s Ž.1,0,y1, 55e s Ž.0,0,0 . Note that the element ce1 is nonzero and generates the socle of NЈ. The 2 2 element c annihilates NЈ and so NЈ is a multigraded RЈ s QЈrŽc .- module. Let K be the Koszul complex on Ä4a, b, c with generators A, B, C. QЈ For a QЈ-module P we identify Tor Ž.P, k with HP ŽmK ., and denote homology classes by clsŽ.᎐ . QЈŽ. Ž. The k-vector space Tor1 RЈ, k is generated by w s cls cC . QЈ On the other hand, Tor2 Ž.NЈ, k is generated by z11s clsŽ.Ž.ce m Ž.A n C , z21s cls ce m Ž.B n C , z33sclsŽ.ce m Ž.A n C . Indeed, a direct computation shows that these elements are cycles. On the Ž. other hand, the boundaries in NЈ m K 2 are elements of the form mmŽ.aB n C y bA n C q cA n B , with m g NЈ. Comparing multide- grees, one easily sees that z123, z , z are linearly independent. Furthermore QЈŽ. 3 Ä4 QЈŽ. Tor23NЈ, k s G m k ( k ; hence z 1, z2, z3is a basis for Tor 2NЈ, k . The action of Tor QЈŽ.RЈ, k on Tor QЈŽ.NЈ, k can be computed from the natural DG module structure of Ž.NЈ m K over the DG algebra Ž.RЈ m K . Using the bases above, we immediately get that QЈ QЈ Tor12Ž.RЈ, k и Tor Ž.ŽNЈ, k s wz 1, wz2, wz3 .Ž.s 0. Thus, to prove that oNŽ.Ј/0, where : QЈ ª RЈ is the canonical QЈ surjection, it is enough to exhibit a nonzero element in KerŽ Tor3 Ž.NЈ, k RЈŽ.. ªTor3 NЈ, k . Ž. Consider the cycle u s ce1 m A n B n C g NЈ m K. Let KЈ s RЈ m Kbe the Koszul complex over RЈ, and let RЈ²:S denote the divided powers algebra, with < RЈ²:S with differential defined by Žn.Žn.Ž< is an RЈ-free resolution of k; cf.wx 27, Theorem 4 . The map Tor QЈŽ.NЈ, k RЈŽ. ªTor NЈ, k is induced by the natural inclusion NЈ mQЈ K s NЈ mRЈ KЈ ª NЈ mRЈ F. A routine computation shows that Ž2. u s Ѩ Ž.e12m Ž.A n B m S q e m S q e3m Ž.A n C m S . Ž. Ž QЈŽ. Hence u is a boundary in NЈ mRЈ F, and cls u g Ker Tor3 NЈ, k ª RЈ Tor3 Ž..NЈ, k . Note that the maximal degree of a generator in the minimal resolution of NЈ is 4. Thus, byw 2,Ž. 1.5g x, we get that oMŽ.Ј/0, where MЈ s 5 NЈrᒊ NЈ. The QЈ-module MЈ has finite length, and it is perfect with pdQЈ M s 3. For completeness, we note that the computer program MACAULAY gives for MЈ the following Betti numbers: 3, 18, 27, 12. Following a method of Avramovw 2,Ž. 2.5x , we useŽ. 2.4 above to prove Ž.2.5 PROPOSITION. Let Ž Q, m, k . be a local ring and x, y, zbeaQ-regu- lar sequence. The Q-module M presented by the 3 = 18 matrix y 0 xz2 000 0 0 0 0 0 0 0 0 0 0 0 0z00 0x5 00x4 y x 32 y x 23 y xy 4 y 5 x 4 z x 3 yz x 22 y z xy 3 z y 4 z ž/zx00z2540yyz00 00000 0 00 2 Ž. is perfect with pdQ M s 3. Furthermore, zMs0and o M / 0, where : 2 Q ª R s QrŽ z. is the natural surjection. Proof. Denote by Q˜ the localization of the polynomial ring Qawx,b,c at the maximal ideal ᒊ q Ž.a, b, c , and by QЈ the localization of kawx,b,c at Ž.a, b, c . Consider the commutative diagram: Q ¤ Q˜ ª QЈ xx x 22 R¤Q˜rŽ.cªQЈr Ž.c where the squares are tensor product diagrams, is factorization by ᒊQ, and is the unique map of Q-algebras with Ž.a s x, Ž.b s y, and Ž.csz. Let M˜˜be the module over Q presented by the 3 = 18 matrix above, with x, y, z substituted by a, b, c. Set M s M˜˜m Q and MЈ s M m 2 QЈ, and note that ÄQЈ, QЈrŽc ., MЈ4 is the triple constructed inŽ. 2.4 . 206 SRIKANTH IYENGAR As the entries in the presentation of M˜ are monomials in the variables, and x, y, z is a Q-regular sequence, it is easily seen that, for i ) 0, QQ˜˜2 ToriiŽ.M˜˜, Q s 0 and Tor Ž.QrŽ.c , Q s 0, Ž.2.5.1 QQ˜˜2 ToriiŽ.M˜˜, QЈ s 0 and Tor Ž.QrŽ.c , QЈ s 0. 2 For the map : QЈ ª QЈrŽ.c , we have oM ŽЈ ./0 by Ž 2.4 . . Applying the flat base change property of the obstruction, cf.w 2,Ž. 1.4x , first to the homomorphism and then to the homomorphism , we deduce that oM Ž./0. It remains to be seen that M is perfect of grade 3. FromŽ. 2.5.1 we get pdQQQM s pd ˜ M˜˜s pd ЈMЈ s 3. From the presentation for M, it is clear that Ž.a555, b , c annihilates M˜, and hence that Ž.x 555, y , z annihilates M. 5 5 5 As x , y , z is a Q-regular sequence, we see that grade M G 3 s pdQ M. Thus, M is perfect. Ž.2.6 Proof of Theorem Ž.2.2 . Suppose that x, y, z is a Q-regular se- 2 quence. Let M and be as inŽ. 2.5 and set f s z . As depth Q G n G 3 and depth M s depth Q y 3, there exists a Q-regular sequence a1,...,any3 which is also M-regular. Ž. Set L s M m Qr a1,...,any3 . Note that L is perfect and also that: Q Q Tor Ž.L, k s Tor ŽM, k .⌳, mk R R Tor Ž.L, k s Tor ŽM, k .⌳, mk ⌳ QŽŽ .. where s Tor Rr a1,...,any3 ,k is the exterior algebra on the vector Ž. Ž. Ž. space ⌳1. By naturality we have oLsoMmk ⌳, and hence 2.5 shows that oL Ž./0. As recalled inŽ. 2.3 above, the nontriviality of the obstruction implies that the minimal Q-resolution Y of L has no DG module structure over the Koszul complex K resolving R. If X is a DG algebra resolving R, then, by the universal property of Koszul complexes, there is a homomorphism of DG algebras K ª X. Thus, if Y is a DG module over X, then Y has the structure of a DG module over K, which is a contradiction. 3. MINIMAL RESOLUTIONS In this section Ž.Q, ᒊ, k is a commutative noetherian local ring, : Q ¸ R is a surjective homomorphism, and M is a finite R-module. We FREE RESOLUTIONS AND RINGS 207 useŽ. 1.2 to construct the minimal R-free resolution of a dth syzygy of M, where d is known a priori. Ž.3.1 . The module M has a minimal R-free resolution Ѩ Ѩ d Ѩ 6 1 иии dy1 иии ª Gddª G y10ª G , which is unique up to an isomorphism. Thus Mdd, the image of Ѩ , is well defined and is called the dth syzygy module of M over R. Ä4 For d ) max depth R y depth M,0 q1 the R-module Md has no free Ž. direct summands; cf.w 3, 4.7xw . Furthermore, depth Md s depth R, by 22, Proposition 10x , and hence pdQdM s pd QR if pd QR and pd QM are finite. Ž. 3.2 THEOREM. Assume that pdQQR s 1 and that pd M is finite. For d G depth R y depth M q 2, the dth syzygy module Md, of M o¨er R, has a minimal free resolution Y with a DG module structure o¨er the Koszul complex X resol¨ing R. Ž. Ž. The free resolution of F X, YofMgid ¨en by Theorem 1.2 is minimal. Remark. The resolution described above is the periodic of period 2 minimal resolution constructed by Shamashwx 24 and Eisenbud wx 10 . Proof. The Koszul complex X:0ªQe ª Q ª 0, where ѨŽ.e s x is a nonzero divisor, is a free resolution of R. It has a DG algebra structure with e и e s 0. 2 Ž. If x f ᒊ , then by a theorem of Nagataw 21, 27.4x we have pdR M s pdQRM y 1; hence pd M s depth R y depth M and Mds 0. 2Ž. Assume x g ᒊ . By 3.1 , we have pdQdM s pd QR and so Mdhas a minimal free resolution nn Y:0ªY10ªY ª0 with Y0( Q and Y1( Q . As Md is an R-module, multiplication by x on Y is homotopic to zero. 2 Choose one such homotopy , and note that for degree reasons s 0. Setting e и y s Ž.y , for y g Y, gives Y the structure of a DG X-module. Žn. With e s e m иии m enŽ.copies , the free resolution F s FXŽ.,Y given by TheoremŽ. 1.2 is Žn. Žn. Žny1. иии ª Re m Y10ª Re m Y ª Re m Y 1ª иии Žn.Žny1.Ž. Žn.Žny1.Ž. with maps e m y0011¬ e m y and e m y ¬ e m y for Žn. yiigY. Setting Y is Re m Yi, the R-free resolution of Mdtakes the form иии ª Y101ª Y ª Y ª иии ª Y 10ª Y , where s R m and s R m . 208 SRIKANTH IYENGAR Ž. As Y is minimal Y10; ᒊY . To show that the periodic resolution is Ž. minimal, we need to check that Y01; ᒊY . This follows from the fact that Md has no free direct summands by an argument of Eisenbudw 10, Ž. Ž. 6.1x , which we reproduce for completion. As F is exact, Y0 ( Ž. Ž. Ž. Coker ( Md.If Y010were not contained in ᒊY , then Y would contain a basis element of Y10,soŽ.Y would have an R-free direct summand. Since Md has not free direct summands, this concludes the proof. Ž. 3.3 . If pdQ R s 2, then by the Hilbert᎐Birch theorem R has a mini- mal free resolution nnq1 X:0ª Qfjiª Qe ª Q ª R ª 0 [[js1is1 Ž. Ž .i with eiisy1a, where a is a nonzero divisor and for I, J ; ގ, the J element I is the minor obtained by deleting the rows indexed by I and columns indexed by J. Herzogwx 14 has shown that the formulas 0ifisj, ¡n aŽ.1iqjqkkfif i - j, eijиes~yÝyi,jk ks1 ¢yejiиe if i ) j define on X a structure of a commutative DG algebra. Ž. 3.4 THEOREM. Assume that pdQQR s 2 and that pd M if finite. Let X be the free resolution of R with the DG algebra structure as abo¨e. For d G depth R y depth M q 2, the dth syzygy module Md, of M o¨er R, has a free resolution Y with a DG module structure o¨er X. Ž. Ž. The free resolution F X, YofMgid ¨en by Theorem 1.2 is minimal if and only if the kernel of : Q ª R is not generated by a regular sequence. Remark. For the case not covered by the theorem, that is, when kerŽ. is generated by a regular sequence, Avramov and Buchweitzwx 5 build tail ends of minimal resolutions using different techniques. Ž. Proof. By the assumption on d and 3.1 , we have depth Md s depth R. Thus the minimal free resolution of Md is Y :0ªY210ªY ªY ª0, and has the structure of a DG module over X byŽ. 2.1 . FREE RESOLUTIONS AND RINGS 209 If kerŽ. is generated by a regular sequence, then X is a Koszul и Ž. Ž . complex and hence XqqX o ᒊ X q. Thus, condition ii of 1.9 fails to hold and FXŽ.,Y cannot be minimal. If kerŽ. is not generated by a regular sequence, then is a Golod homorphism; cf.w 1,Ž. 7.1x . As the minimal resolutions X and Y have the appropriate DG structures, if Md is -Golod, then FXŽ.,Y is minimal by Ž.1.9 . To prove that Md is -Golod, it is enough, byw 19,Ž. 1.1x , to show that QŽ. RŽ. ToridM , k ª Tor idM , k is injective for i G 1. Lescotw 18, Lemma, p. QŽ. 43x proves that if is a Golod homomorphism and Tori M, k ª RŽ. QŽ.R Tori M, k is injective for all i G p, then ToripM y1, k ª Tori Ž. Mpy1,kis injective for all i G 1. As pdQ M is finite, this applies to p s pdQ M q 1 s d q 1. 22 Ž.3.5 EXAMPLE. Let Q s kxwx,y for a field k. Set R s QrŽxy,xy . 22 and M s RrŽ x , y .. We have depth M s depth R s 0. Thus d s 2, and X Ž2 . the second syzygy module is the direct sum of M2 s kxwx,yrx,xy and Y Ž2 . X M2 skxwx,yry,xy . The Q-free resolution of M2 is yy 2 ž/x6 Ž.xxy6 Y:0ªQc Qb12[ Qb Qa ª 0. Ž. By 3.3 , the multiplication on X is defined by e12и e syxyf sye21иe . Following through the proof ofŽ. 2.1 , one sees that the action of X on Y is given by e111222и a s yb , e и b s xyc, e и a s yb , e21иbsyxyc, f и a s yc. Now, TheoremŽ. 1.2 gives the beginning of the resolution of kxwx,yr Žx2,xy.over R: yy 0 xy yxy 0 2 yxxy 00 yyy0 ž/x00 xxy2 x0y 2 y 6 ž/6 Ž.xxy6 532 иии ª RRRR. 2 The resolution of kxwx,yrŽy,xy.is obtained by interchanging x and y. 2 2 1 The Poincare´ series of M isŽ 1 q t y t .Ž1 y t y t .y . 4. MONOMIAL QUOTIENTS OF MONOMIAL RINGS Let k be a field and Q s kxwx1,..., xn the polynomial ring with the natural n-grading, and unique homogeneous maximal ideal ᒊ s 210 SRIKANTH IYENGAR Ž.x1,..., xn .If ᑾis an ideal generated by monomials Ž.a1,...,as , then the quotient ring R s Qrᑾ is n-graded with the multigrading induced from Q. Ž. ⌳ s 4.1 Taylor resolutions. Define X to be the graded Q-module [is1 Qf , with X free of rankŽ.s . For I Ä4i ,...,i set f f иии f and ip p s 1pIis 1n n ip aI to be the least common multiple of the monomials indexed by I. The Taylor resolution, cf.wx 28 , of R over Q is the graded module X with differential p k aI Ѩ Ž.fI syÝ Ž1 . fÄIi4. a _k ks1 ÄI_ik4 As proved by Gemedawx 13 , cf. also wx 12 , the formula f и f gcdŽ.a , af f II12s III121n I 2 defines a structure of a DG algebra on the Taylor resolution. Ž.4.2 Multihomogeneous quotients. Let ᑿ be an ideal generated by monomials b1,...,bt , and such that ᑾ : ᑿ. Let Y be the Taylor resolution of Q ᑿ and h h иии h , where J Ä4j ,..., j . r Jjs 1n n jq s 1 q For each i there is an integer Ž.i such that b Ži. divides ai. For I Ä4i,...,i , set Ž.I Ä Ž.i ,..., Ž.i 4and h h иии h s 1 p s 1 p Ž I . s Ži1. n n Ži p. in Y. The consideration of such ‘‘monomials’’ is suggested by the formulas for the differential in the resolution of the residue field given inwx 9 . A direct computation shows that LEMMA. The canonical surjection R s Qrᑾ ¸ Qrᑿ extends to a homo- morphism, : X ª Y, of DG Q-algebras defined by aI Ž.fI s hŽI.. bŽI. In particular, Y is a DG X-module. As an application ofŽ. 1.2 and Ž. 4.2 , we get an explicit free resolution of the R-module M s Qrᑿ. Ž.4.3 THEOREM. With X and Y as abo¨e, set X s XrQ. There is a projecti¨e resolution of M o¨er R, which in degree n is the R-module R m Xiijm иии m X m Y [ 1 p pqi1qиии qi pqjsn FREE RESOLUTIONS AND RINGS 211 and differential Ѩ s Ѩ Ј q Ѩ Љ, where Ѩ Ј f иии f h Ž.IIJ1m m pm p e Ž.1 iy1f иии Ѩ f иии f h syÝ y IX1m m Ž.Iipm m IJm is1 e Ž.1 pf иии f Ѩ Ž.h , qy IIYJ1m m pm ѨЉfиии f h Ž.IIJ1mm pm py1 e Ž.1i gcd Ža , af .Ž.иии f f иии f h syÝII12 I 1mm Iiin Iq1m m IJpm is1 ab IJp 1qepy1 qyŽ.1 Ž.fIImиии m f m hŽI.n h J b1 py1p ÄŽIp.jJ4 Ž. Ž. with ei s i q card I1 q иии qcard Ii . We have the following combinatorial criterion for the minimality of the resolution: Ž.4.4 PROPOSITION. The resolution of Theorem Ž.4.3 is minimal if and only if the following conditions hold: Ž. Ä4 1aI /aÄI_i4 for 1 F i F s, where I s 1,...,s . Ž. Ä4 2bJ /bÄJ_ j4 for 1 F j F t, where J s 1,...,t . Ž. Ž . 3 gcd aij, a / 1 for 1 F i, j F s. Ž. Ä4 4aI /bŽI. for all I : 1,...,s . Proof. Let I12and I be subsets ofÄ4 1, . . . , s , and J a subset of Ä41,...,t . Evidently, the resolution is minimal if and only if X and Y satisfy the following conditions: Ž.1Ј Xis a minimal resolution. Ž.2Ј Yis a minimal resolution. Ž.3Ј gcd Ža , a ./ 1. II12 Ž.4Ј ab/b . IJppÄŽI.jJ4 A result of Froberg¨ w 12,Ž. 5.1x shows that conditionsŽ. 1 and Ž 1Ј ., respectively,Ž. 2 and Ž 2Ј ., are equivalent. Assuming thatŽ. 1 holds, a direct computation shows thatŽ. 3 and Ž 3Ј ., respectively,Ž. 4 and Ž 4Ј ., are equivalent. As a corollary ofŽ. 4.3 and Ž. 4.4 , we recover the main result ofwx 9 . 212 SRIKANTH IYENGAR COROLLARY. Let Q s kxwx1,..., xn be a polynomial ring o¨er a field k, and let ᑾ be an ideal minimally generated by monomials a1,...,as. With X Ä4 the Taylor resolution of R s Qrᑾ and Y the Koszul complex on x1,..., xn , Theorem Ž.4.3 yields a free resolution of the residue field o¨er R. Ž. Ä4 Furthermore, if aII/ a_iij and gcd a , a / 1 for i, j g I s 1,...,s , then the resolution is minimal. Proof. When M s k, the Taylor resolution is the Koszul complex on the variables Ä4x1,..., xn . Thus, with X and Y as above,Ž. 4.3 is an R-free resolution of the residue field k. Under the assumptions of the corollary, conditionsŽ. 1 and Ž. 3 of Ž. Ž .2 Ž. Ž. Proposition 4.4 are satisfied. As ᑾ ; x1,..., xn , conditions 2 and 4 are easily seen to hold. Thus, the resolution is minimal. Remark. Recently, Backelinwx 7 and Eisenbud, Reeves, and Totarow 11 x have given iterative procedures for constructing a resolution of a multi- graded module over a monomial ring. Their purpose is to give upper bounds on the degrees of the generators in the minimal resolution, in terms of the degrees of the generators of the first syzygy module. Our aim in this paper has been to give explicit resolutions, and the price we pay is that the upper bounds we get from TheoremŽ. 4.3 are weaker. Remark. 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