The Theory of L-Complexes and Weak Liftings of Complexes

188, 144]183 Ž1997. JA966821

JOURNAL OF ALGEBRA ARTICLE NO.

The Theory of L-Complexes and Weak Liftings of Complexes Yuji Yoshino Institute of Mathematics, Faculty of Integrated Human Studies, Kyoto Uni¨ ersity, Kyoto 606-01, Japan Communicated by Mel¨ in Hochster Received April 24, 1994

INTRODUCTION The notion of a weak lifting of a finite module is introduced and studied intensively by Auslander et al. in w2x, and it turned out that this notion is well applied to the theory of Cohen]Macaulay approximations. The present paper is devoted to studying such lifting problems in the framework of complexes, which will be done by introducing the notion of an L-complex. The lifting and the weak lifting of a complex are defined in the following way. For a commutative ring R, we denote by Kqf g Ž R . the category of complexes consisting of finite free R-modules and that are bounded below. Let S ª R be a ring homomorphism of commutative rings and let F be a complex in Kqf g Ž R .. Then the complex G in Kqf g Ž S . is called a lifting of F to S if there is a chain isomorphism G mS R ( F . The complex F is said to be liftable to S, when such a lifting G g Kqf g Ž S . exists. If F is only a direct summand of G mS R for some G g Kqf g Ž S ., then G is a weak lifting of F to S and F is said to be weakly liftable to S. We are interested distinctively in the case when R is isomorphic to SrI where I is an ideal of S generated by a regular sequence. Under the circumstances, if F is an R-free resolution of a finite R-module, then it can be seen that F is weakly liftable Žresp. liftable . to S if and only if M is weakly liftable Žresp. liftable . in the sense of w2x. There are two reasons why we extend the notion to the category of free complexes. First, as in w11x, the classification of particular modules over a commutative ring is sometimes reduced to the classification of representav

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144 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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tions of an Artin algebra through certain isomorphisms of complexes. It is, thus, thought to be natural to consider the category of complexes even if we focus on modules. Second, as we develop below in this paper, every argument concerning the lifting and the weak lifting becomes much simpler for complexes than for modules. Indeed, we shall introduce the new notion that we call an L-complex ŽL for lifting., and using this, we can give a unified method to treat the lifting and the weak lifting of a complex. Furthermore the L-complex will turn out to be a very natural object, and it has some good applications, such as, for constructing the Eisenbud resolution of a complex, or for the virtual projective dimension for a complex. Thus the main purpose of this paper is to introduce the notion of an L-complex and to apply it to several cases including the problem of weak liftings. Now we shall give a brief description of the organization of this paper. In Section 1, we present basic notions such as a lifting and a weak lifting of a complex, and we show briefly the conditions for a complex to be liftable. The reader should notice that the Eisenbud operators will play a central role in showing these conditions. We introduce the notion of an L-complex in Section 2. To be more precise, let us consider the ring R s SrxS, where S is a local ring and x is a non-zero divisor. Given a complex F over R in Kqf g Ž R ., we construct the L-complex LS ª R Ž F . in Kqf g Ž S .. See Ž2.1.. The uniqueness of the L-complex will be shown and that there is a quasi-isomorphism LS ª R Ž F . ª F . The main result of Section 2 is Theorem Ž2.7., which gives the conditions for a complex to be weakly liftable, in terms of L-complexes. In Section 3, we consider the ring R s TrŽ x 1 , x 2 , . . . , x r .T, where T is a local ring and  x 1 , x 2 , . . . , x r 4 is a regular sequence of T. In this case, for a complex F in Kqf g Ž R ., we can make successive use of constructing L-complexes with respect to the sequence of ring homomorphisms v

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T s T0 ª T1 s Trx 1T ª T2 s Tr Ž x 1 , x 2 . T ª ??? ª Tr s R, and we get a free complex LT ª R Ž F . in Kqf g ŽT .. We show in Ž3.9. that the complex LT ª R Ž F . is uniquely determined only by F , T, and the ideal Ž x 1 , x 2 , . . . , x r .T. In Theorem Ž3.11., we also give some conditions, in terms of LT ª R Ž F ., for F to be weakly liftable to T, which sharpens the result of w2, Proposition Ž3.6.x. As one of the applications of an L-complex, we show in Section 4 that the Maranda theorem holds for complexes. And this generalizes some theorems of Ding and Solberg w5x. In Section 5 we will show how to construct the Eisenbud resolution for any given complex. To explain this, let R be SrxS where x is a non-zero divisor of a local ring S. Suppose that we are given an S-free resolution G v

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of a finite R-module M. Then Eisenbud w6x can construct an R-free resolution of M from G . We will show that this procedure of construction works even if G is a non-acyclic complex, and we get a complex F in Kqf g Ž R . that is quasi-isomorphic to G . We say that the complex F constructed in this manner is the Eisenbud resolution of G . We shall also give its application in Section 5. It is actually applied to the Cohen]Macaulay approximations, and we can reproduce a result of Ding w4x. The last section, Section 6, is devoted to studying the virtual projective dimension and the complexity for a complex. These invariants are defined for modules by Avramov w3x, but naturally generalized to any complexes. We shall show in Theorem Ž6.9. that Avramov’s equality w3, Theorem Ž3.5.x is valid also for any bounded complexes. Through the duality theorem, our result Ž6.9. can be applied to get the equality concerning the virtual injective dimension in Ž6.10.. v

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1. LIFTINGS AND EISENBUD OPERATORS Notation Ž1.1.. Let R be a ring. For a given complex F over R we denote it by v

­nq1

­n

­ny1

??? ª Fnq 1 ª Fn ª Fny1 ª Fny2 ª ??? . ŽWe write ­nF for ­n if some specification is necessary and also write Ž F , ­ F . for this complex.. Let F and G be complexes over R. We say that the set f s  f n4 is a homomorphism from F to G of degree m if each f n is an R-homomorphism Fn ª Gnqm . A chain homomorphism from F to G of degree m is a homomorphism f of degree m such that G f ny 1 ­nF s ­nqm f n for any n. ŽFor the last equality we often write f­ F s ­ G f if it causes no confusion.. By definition, two chain homomorphisms f and g from F to G of degree m are homotopy equivalent if there is a homomorphism h of degree m q 1 with f y g s ­ G h q h ­ F. For a homomorphism f s  f n4 we denote the homomorphism Žy1. n f n4 by f s. Note that f s is a chain homomorphism if and only if f satisfies ­ G f q f­ F s 0. For an integer m, we denote the shifting complex by F w m x, that is, the nth part of F w m x is Fmq n . Note that a chain homomorphism F ª G w m x of degree 0 is a chain homomorphism F ª G of degree m. We are concerned with chain complexes F with each Fn being a finite free module and with Fn s 0 for n < 0, and the category of such complexes over R is denoted by Kqf g Ž R .. So the objects of Kqf g Ž R . are complexes of finite free modules over R which are bounded below, while the morphisms are the chain homomorphisms of degree 0 between complexes. v

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Let F and G be free complexes in Kqf g Ž R . and let n be an integer. The R-module Ext Rn Ž F , G . is, by definition, the set of homotopy equivalence classes of chain homomorphisms in Hom Kqf g Ž R.Ž F , G wyn x.. v

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DEFINITION Ž1.2.. Let S ª R be a ring homomorphism of Noetherian commutative rings. Ža. A free complex F g Kqf g Ž R . is called liftable to S if there is a complex G g Kqf g Ž S . such that there is a chain isomorphism G mS R ( F in Kqf g Ž R .. In this case G is called a lifting of F to S. v

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Žb. A complex F g Kqf g Ž R . is called weakly liftable to S if there is a complex G g Kqf g Ž S . such that F is a direct summand of G mS R; equivalently F [ F X is liftable to S for some F X g Kqf g Ž R .. v

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In the rest of the paper, S always denotes a commutative Noetherian local ring with maximal ideal m and with residue field k s Srm. Further we denote R s SrxS, where x g m is a non-zero divisor on S. Remark Ž1.3.. Ža. According to Auslander et al. w2x, an R-module M is said to be liftable to S if there is an S-module N such that N mS R ( M and TornS Ž N, R . s 0 for n G 1. And similarly M is called weakly liftable to S if there is an R-module M9 such that M [ M9 is liftable. Let F be an R-free resolution of a finitely generated R-module M. Then it is obvious that M is liftable if and only if the complex F is liftable in our sense. Further if M is weakly liftable to S, then the complex F is weakly liftable. In fact, if M [ M9 is liftable, then denoting an R-free resolution of M9 by F X , we can see that F [ F X is liftable. However, note that it is not obvious to see the validity of the converse. When there is a complex F X with F [ F X being liftable, we have to show that F X can be taken as a free resolution of some R-module, that is, F X is acyclic. And this will be proved in Ž2.7.. v

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˜ be an S-homomorphism of S-free modules. AsŽb. Let f˜: F˜ ª G ˜ ˜ Then, since x is a sume that f mS R s 0 or equivalently Im f˜: xG. ˜ ˜ ˜ ª G˜ by non-zero divisor on G, f passes through the multiplication map G ˜ ˜ x; that is, there is an S-module homomorphism ˜ g: F ª G such that f˜s xg. ˜ In case that f˜ is a homomorphism F˜ ª G˜ between free complexes over S, the same is true. And if f˜ is a chain homomorphism, then ˜ g ˜ is a chain is also a chain homomorphism. In particular, if f˜: F˜ ª G homomorphism and if f s f˜mS R is homotopy equivalent to the zero map, then we can find homomorphisms ˜ h and u ˜ satisfying the equality f˜s ˜h ­ F˜ ˜˜ G q ­ h q xu. ˜ This will be often used without any comment in the remainder of the paper. v

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We now recall some of the fundamental notions from the paper of Eisenbud w6x. For a complex F g Kqf g Ž R . we can lift the maps ­n to S-linear maps ­˜n and can construct a sequence F˜ of S-linear maps: v

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­˜nq1

­˜n

­˜ny1

??? ª F˜nq 1 ª F˜n ª F˜ny1 ª F˜ny2 ª ??? .

Ž 1.4.1.

Note that each F˜n is taken as a free module over S and ­˜n mS R s ­n . Note also that F˜ may not be a complex. However, for each n, since ­ny 1 ­n s 0 and since x is a non-zero divisor on S, there is an S-module homomorphism ˜t n : F˜n ª F˜ny2 with ­˜ny1 ­˜n s xt˜n . We denote t n s ˜t n mS R. The homomorphisms t s  t n4 and ˜t s ˜t n4 are called the Eisenbud operators of F . Because of the equality x ­˜˜t s ­˜3 s xt˜­˜, we see that the operator ˜t is commutative with ­˜. Hence t: F ª F is a chain homomorphism of degree y2. Furthermore we can prove that t is uniquely determined Žindependently of the choice of ­˜. up to chain homotopy of F . In fact, if ­˜9 is another ˜ then we can write ­˜y ­˜9 s xh˜ for some ˜h, lifting of ­ and if Ž ­˜9. 2 s xt9, since Ž ­˜y ­˜9. mS R s 0 and since x is a non-zero divisor on S. Thus we have v

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˜. xt˜s ­˜2 s Ž ­˜9 q xh

2

˜ q ˜h ­˜9 q xh˜2 . . s x Ž ˜t9 q ­˜9h ˜2 , and setting h s ˜h mS R, we have Therefore ˜t y ˜t9 s ­˜9h q ˜ h ­˜9 q xh t y t9 s ­ h q h ­ as desired. See also w6x. Thus the following is well defined. DEFINITION Ž1.4.. For a complex F g Kqf g Ž R ., the Eisenbud class e F of F is the homotopy equivalence class in Ext 2R Ž F , F . of the chain homomorphism t. v

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The next three lemmas may be well known. Compare with w2, Propositions Ž1.5., Ž1.6., and Ž2.5.x. LEMMA Ž1.5.. only if e F s 0.

For a complex F g Kqf g Ž R ., F is liftable to Srx 2 S if and v

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Proof. Suppose e F s 0. Then it follows from the definition that we can ˜ we find homomorphisms ˜ h and ˜ g with ˜t s ­˜˜ hq˜ h ­˜q xg. ˜ Since ­˜2 s xt, 2 2 2 ˜. s x Ž ˜g q ˜h .. Replacing ­˜ by ­˜y xh, ˜ we may assume from have Ž ­˜y xh ˜ Thus G s F˜ mS Srx 2 S is actually a complex, and it this that Im ­˜2 : x 2 F. satisfies G mS r x 2 S R s F . Therefore F is liftable to Srx 2 S. The converse is proved similarly. v

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AND WEAK LIFTINGS

LEMMA Ž1.6.. Assume that S is a complete local ring. If Ext R2 Ž F , F . s 0 for a complex F g Kqf g Ž R ., then F is liftable to S. v

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Proof. We shall show that for a given n G 2, the lifting ­˜ can be chosen so that Im ­˜2 : x n F˜ . Since e F s 0 by the assumption, this is true for n s 2 by the proof of Ž1.5.. If it is true for n, then we can write ­˜2 s x n˜t Ž n. for some homomorphism ˜t Ž n. of degree y2. By the same reason by which t s ˜t mS R is a chain homomorphism, we see that ˜t Ž n. mS R is a chain homomorphism of degree y2; hence it is homotopy equivalent to zero by the assumption. Thus we have homomorphisms ˜ s Ž n. and u ˜ of Ž n. Ž n. degree, respectively, y1 and y2 such that ˜t s ­˜˜ s q˜ s Ž n.­˜q xu. ˜ It follows from this that v

2

Im Ž ­˜y x n˜ s Ž n. . : x nq1 F˜ . v

Starting from n s 2, we continue this procedure to construct a homomorphism

˜ s ­˜y D

`

Ý x n˜s Ž n. , ns2

which converges since S is complete. Clearly from the construction, it ˜2 s 0 and D ˜ mS R s ­ ; hence Ž F˜ , D ˜ . is a complex that is a satisfies D lifting of F . v

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LEMMA Ž1.7.. Assume that S is a complete local ring and that F g Kqf g Ž R . is liftable to S. If Ext 1R Ž F , F . s 0, then the lifting of F is unique up to chain isomorphism. v

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Proof. Suppose that two complexes Ž F˜ , ­˜. and Ž F˜ , ­˜9. in Kqf g Ž S . are liftings of F . We want to construct a chain isomorphism between them. First we assume that ImŽ ­˜y ­˜9. : x n F˜ . ŽNote that this is true for n s 1, since both ­˜ and ­˜9 are liftings of ­ .. We are then able to find ˜ h such that ­˜9 s ­˜q x n ˜ h, for x is a non-zero divisor on S. Since ­˜2 s ­˜9 2 s 0, we have ­˜˜ hq˜ h ­˜q x n ˜ h 2 s 0; hence putting h s ˜ h mS R we see that ­ h q h ­ s 0. This shows that h s is a chain homomorphism of degree y1. ŽSee Ž1.1... Since Ext 1R Ž F , F . s 0, there is a homomorphism g of degree 0 with h s s ­ g q g ­ . Taking a lifting homomorphism ˜ g of g, we can find ˜ This a homomorphism f˜ of degree 0 on F˜ such that ˜ h s s ­˜˜ gq˜ g ­˜9 q xf. s s˜ s ˜ ˜ ˜ shows that h s ­ ˜ g y˜ g ­ 9 q xf ; hence v

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Ž 1 q x n ˜g s . ­˜9 y ­˜Ž 1 q x n ˜g s . s Ž ­˜9 y ­˜. q x n ž ˜g s­˜9 y ­˜˜g s / s x nq 1 f˜s .

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YUJI YOSHINO

Denoting u ˜Ž n. s 1 q x n ˜g s , we have, in particular, ImŽ uŽ n. ­˜9 y ­˜uŽ n. . : nq 1 ˜ x F . Note that u ˜Ž n. may not be a chain homomorphism, but it gives an automorphism on each piece of F ; hence we may write ImŽ uŽ n. ­˜9uy1 Ž n. y ­˜. : x nq 1 F˜ . Starting from n s 1, we obtain the sequence uŽ1. , uŽ2. , uŽ3. , . . . of homomorphisms on F˜ that satisfy v

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Im Ž uŽ n. uŽ ny1. ??? uŽ1. . ­˜9 Ž uŽ n. uŽ ny1. ??? uŽ1. .

ž

y1

y ­˜ : x nq1 F˜ .

/

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Since S is complete, we see that the product Ł`is1 uŽ i. converges to a homomorphism u ˜ on F˜ that is an isomorphism on each piece of F˜ . Then, by the above, we have u ˜­˜9 s ­˜u; ˜ hence u˜ gives a chain isomorphism between Ž F˜ , ­˜. and Ž F˜ , ­˜9.. v

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2. L-COMPLEXES As in the previous section let S be a Noetherian local ring with maximal ideal m, and R s SrxS where x is a non-zero divisor on S. Now we would like to consider the condition for a complex to be weakly liftable. To do this we propose a new notion that is naturally defined as follows: DEFINITION Ž2.1.. Let F be a complex in Kqf g Ž R ., and let Ž F˜ , ­˜. be as in Ž1.4.1.. Further ˜t is the Eisenbud operator given by ­˜. Then the L-complex L s LS ª R Ž F . in Kqf g Ž S . is defined by v

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L n s F˜ny1 [ F˜n ,

­nL s

ž

­˜ny 1

yt˜n

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y­˜n

/

.

L ˜L Sª RŽ It is easy to see that ­˜ny F . is actually a complex. 1 ­n s 0; hence L Throughout this section, since the ring homomorphism S ª R is always fixed, we simply write L Ž F . for LS ª R Ž F .. v

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Definition Ž2.1. might give the reader the impression that L Ž F . depends on the choice of ­˜, but we can prove in Lemma Ž2.2. that it is independent. v

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LEMMA Ž2.2.. The L-complex L Ž F . is uniquely determined up to chain isomorphism, so it is independent of the choice of Ž F˜ , ­˜.. v

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Proof. Let ­˜9 be another lifting of ­ and let ˜t9 be the corresponding Eisenbud operator. Then there is a homomorphism ˜ h of degree y1 such

L-COMPLEXES

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˜ Thus we have that ­˜9 y ­˜s xh. 2

2

˜ s Ž ­˜9 . s Ž ­˜q xh˜. s x Ž ˜t q ­˜˜h q ˜h ­˜q xh˜2 . ; xt9 ˜2 . This shows the following equality of hence ˜t9 s ˜t q Ž ­˜˜ hq˜ h ­˜. q xh matrices:

ž ˜ / ž ˜ ˜˜/ ž ­ x

h 1

1 0

yt y­

˜ s ­˜9 yh 1 x

1 0

˜ yt9 . y­˜9

/ ž

/

Thus the matrix Ž 10 ˜h1 . gives a chain isomorphism between the two Lcomplexes defined through ­˜ and ­˜9. LEMMA Ž2.3.. There is a chain homomorphism c : L Ž F . ª F that is a quasi-isomorphism. In particular, L Ž F . is isomorphic to F in the deri¨ ed category Dqf g Ž S .. v

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Proof. Let pn : F˜n ª Fn be a natural projection for any n and consider p s  pn4 . Then it is easy to see that the maps cn s Ž0, ypn .: F˜ny1 [ F˜n ª Fn yield a chain homomorphism c : L Ž F . ª F . On the other hand, consider the complex LX defined as follows: v

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LXn s F˜ny1 [ F˜n , X

ž

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­˜ny 1

yxt˜n

1

y­˜n

/

.

Note that LX is a split exact sequence, so it has trivial homologies. Then we define a homomorphism f s Ž 10 0x . from LX to L Ž F ., and we can verify the equality v

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yt˜ y­˜

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0 1 s x 0

0 x

/ž / ž /ž

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yxt˜ . y­˜

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0 ª LX ª L Ž F . ª F ª 0 v

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is an exact sequence of complexes. Since L has only trivial homologies, c is a quasi-isomorphism of complexes as desired. v

Remark Ž2.4.. Ža. Let F be an R-free resolution of a finitely generated R-module M. Then it follows from Ž2.3. that L Ž F . is an S-free resolution of M. But even if F is a minimal complex Ži.e., ­ mR k s 0., L Ž F . may not be a minimal complex. This occurs when ˜t mS k / 0. v

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YUJI YOSHINO

Žb. The complex L Ž F . mS R is nothing but the mapping cone of the Eisenbud operator t. In fact, we have v

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ž

­ w y1 x 0

yt . y­

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DEFINITION Ž2.5.. Let F be a free complex in Kqf g Ž R .. We define the initial index iŽ F . of F as follows: v

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i Ž F . s inf  i N Hi Ž F . / 0 4 . v

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Note that if F is a minimal complex Ži.e., ­ F mR k s 0., then F has the form v

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??? ª FiŽ F v .q2 ª FiŽ F v .q1 ª FiŽ F v . ª 0. For an integer m we can define the truncated complex tmŽ F . at m as v

tm Ž F . n s v

­nt m Ž F v . s

½ ½

Ž if n G m . , Ž if n - m . ,

Fn 0

­nF

Ž if n ) m . , Ž if n F m . .

0

The Žfirst. syzygy complex V F is, by definition, the complex Žt iŽ F v .q1 F . w1x. v

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Remark Ž2.6.. Ža. Let F be a free resolution of a finitely generated R-module M. If F is minimal, then V F is the minimal free resolution of the first syzygy module V R Ž M .. While, in case that F may not be minimal, V F is a free resolution of a direct sum of V R Ž M . with a free module. Žb. We may regard a free module G as a complex G in Kqf g Ž R . that is defined by v

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Gn s

½

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Ž if n s 0 . , ­ G s 0 Ž for all n . . Ž otherwise . , n

Under this notation, it can be easily checked that there is an exact sequence of complexes 0 ª FiŽ F v . yi Ž F . ª F ª V F w y1 x ª 0. v

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Now we can state our main result in this section. THEOREM Ž2.7.. For a minimal free complex F in Kqf g Ž R ., the following conditions are equi¨ alent. v

Ža. F is weakly liftable to S. Žb. e F s 0 in Ext 2R Ž F , F .. Žc. F is liftable to Srx 2 S. Žd. There is a chain isomorphism L Ž F . mS R ( F wy1x [ F . Že. There is a chain isomorphism V L Ž F . mS R ( F [ V F . v

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Remark that if F is a minimal R-free resolution of an R-module M, then the abo¨ e conditions for F are equi¨ alent to V S Ž M . mS R ( M [ V R Ž M .. v

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Proof. The equivalence of Žb. and Žc. has been shown in Ž1.5.. The implications Žd. « Ža. and Že. « Ža. follow from the definition of weak lifting. Žb. « Žd.: If e F s 0, then as in the proof of Ž1.5. we can take ­˜ as Im ­˜: x 2 F˜ ; hence t s ˜t mS R s 0. Then we have ­ L v Ž F v . mS R s Ž ­0 y0­ ., which shows that L Ž F . mS R s F wy1x [ F . Žd. « Že.: Before proving this, we note that L Ž F . is a minimal complex, since the right-hand side in Žd. is a minimal complex. Furthermore since L Ž F .rxL Ž F . ( F wy1x [ F , it follows from Nakayama’s lemma that iŽ L Ž F .. s iŽ F .. Thus setting i s iŽ F ., we have isomorphisms: v

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V L Ž F . mS R ( t iq1 L Ž F . w 1 x mS R v

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( t iq1 Ž L Ž F . mS R . w 1 x v

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( t iq1 Ž F w y1 x [ F . w 1 x v

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( Ž F w y1 x [ t iq1 F . w 1 x v

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( F [ VF . v

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Ža. « Žb.: Let G be a complex in Kqf g Ž S . that satisfies G mS R ( F [ F for some complex Ž F X , ­ 9. in Kqf g Ž R .. By this isomorphism we see that each Gn is a lifting free module of Fn [ FnX and ­nG mS R s Ž ­0 ­09 .. Thus we can write X

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­nGs

­˜n

xu ˜n

x¨˜n

­˜nX

/

X Gny1 s F˜ny1 [ F˜ny1

6

Gn s F˜n [ F˜nX

ž

v

154

YUJI YOSHINO

for some lifting homomorphisms ­˜ and ­˜9 of ­ and ­ 9, and for some homomorphisms u ˜ and ¨˜ of degree y1. Then since Ž ­ G . 2 s 0, we have

ž

­˜ x¨˜

2

˜t q xu˜˜ ¨ ­˜u ˜ q u˜­˜9 . ­˜9¨˜ q ¨˜­˜ ˜t9 q x¨˜˜ u This equality shows that ˜t s yxu ¨ . Therefore t s ˜ t mS R s 0; hence e F s ˜˜ 0s

xu ˜ ˜ ­9

/ ž

/

sx

0 as desired. For the last assertion of the theorem, let F be a minimal free resolution of an R-module M. Assume first that conditions Ža. to Že. hold for F . Then, by Ž2.4.Ža., L Ž F . is an S-free resolution of M. But as shown in the proof of Žd. « Že., L Ž F . is a minimal complex; hence it follows from Ž2.6.Ža. that V L Ž F . is a minimal free resolution of an S-module V S Ž M .. Since x is a non-zero divisor on V S Ž M ., it follows that V L Ž F . mS R is a minimal R-free resolution of V S Ž M . mS R. Thus the isomorphism of complexes in Že. gives an isomorphism of modules V S Ž M . mS R ( M [ V R Ž M .. Conversely, assume that V S Ž M . mS R ( M [ V R Ž M .. Then it follows from Ž1.3.Ža. that F is weakly liftable to S; hence condition Ža. holds. v

v

v

v

v

v

v

v

v

v

v

3. SUCCESSIVE L-COMPLEXES In this section we shall generalize the argument in the previous section to the rings of complete intersection. For this purpose let T be a Noetherian local ring, and let x s x 1 , x 2 , . . . , x r be a regular sequence on T. Furthermore we denote I s xT and R s TrI. Recall first the Eisenbud operators in the present case Žcf. Eisenbud w6x.. Let Ž F , ­ . be a free complex in Kqf g Ž R .. Then we take liftings ­˜n of ­n to T Ži.e., ­˜n mT R s ­n . and consider the sequence of T-homomorphisms: v

­˜nq1

­˜n

­˜ny1

??? ª F˜nq 1 ª F˜n ª F˜ny1 ª F˜ny2 ª ??? . Since ­˜2 mT R s 0, we find the homomorphisms ˜t Ž i. Ž1 F i F r . of degree y2 on F˜ such that ­˜2 s Ý ris1 x i˜t Ž i.. We write t Ž i. for ˜t Ž i. mT R. As before we call ˜t Ž i. and t Ž i. Ž1 F i F r . the Eisenbud operators of F . Noting that IrI 2 is a free R-module having the classes of x 1 , x 2 , . . . , x r as the base, we can see from the same discussion as in Section 1 that t Ž i. Ž1 F i F r . are chain homomorphisms of degree y2 on F , and that each t Ž i. is unique up to chain homotopy. See Eisenbud w6x. Thus we can define the Eisenbud class e FŽ i. ŽT ª R . as the homotopy equivalence class of t Ž i. in Ext 2R Ž F , F .. Note that ˜t Ž i., ˜t Ž j., and ­˜ may not be commutative with one another. Compare with Section 1. v

v

v

v

v

L-COMPLEXES

155

AND WEAK LIFTINGS

As in Section 1 we can prove the following lemmas whose proof will be obtained only by a slight modification of that of Lemmas Ž1.5., Ž1.6., and Ž1.7., and we leave their proof to the reader. LEMMA Ž3.1.. A free complex F in Kqf g Ž R . is liftable to TrI 2 if and only if all the Eisenbud classes e FŽ i. ŽT ª R . Ž1 F i F r . are tri¨ ial in Ext 2R Ž F , F .. v

v

v

LEMMA Ž3.2.. Assume that T is a complete local ring, and that Ext 2R Ž F , F . s 0 for a complex F g Kqf g Ž R .. Then F is liftable to T. v

v

v

v

LEMMA Ž3.3.. Assume that T is a complete local ring, and that a complex F g Kqf g Ž R . is liftable to T. If Ext 1R Ž F , F . s 0, then the lifting of F is unique up to chain isomorphism. v

v

v

v

DEFINITION Ž3.4.. Let F be a free complex in Kqf g Ž R ., and consider the sequence of the local rings: v

Ž 3.4.1. T s T0 ª T1 s Trx 1T ª T2 s Tr Ž x 1 , x 2 . T ª ??? ª Tr s R. Since each Ti is the residue ring of Tiy1 by a non-zero divisor x i , we can construct the L-complex successively Žcf. Ž2.1.. and define LT ª R Ž F . s LT 0 ª T1 LT1 ª T 2 ??? LT ry 1 ª T r Ž F . . v

v

v

v

v

v

LEMMA Ž3.5.. When the sequence Ž3.4.1. is fixed, the L-complex LT ª R Ž F . is unique up to chain isomorphism. v

v

Proof. By virtue of Ž2.2., it is enough to show the following lemma. LEMMA Ž3.6.. Let R s SrxS with x a non-zero di¨ isor on S, and let Ž F , ­ . and Ž F X , ­ 9. be free complexes in Kqf g Ž R .. Assume that there is a chain isomorphism between F and F X . Then LS ª R Ž F . is chain-isomorphic to LS ª R Ž F X .. v

v

v

v

v

v

v

v

Proof. To prove this, let f : F ª F X be a chain isomorphism and let ­˜, ­˜9, and f˜ be lifting maps to S of ­ , ­ 9, and f , respectively. And write ˜ as before. Since ­ 9 s f­fy1, we can find a homo­˜2 s xt˜ and ­˜9 2 s xt9 X ˜˜˜y1 s xu. morphism u ˜ on F˜ of degree y1 such that ­˜9 y f­f ˜ See Ž1.3.Žb.. Thus we have v

v

v

˜ s ­˜92 xt9 ˜˜˜y1 q xu˜ s f­f

ž

2

/

˜˜2f˜y1 q x f­f ˜˜˜y1 u˜ q xu˜f­f ˜˜˜y1 q x 2 u˜2 . s f­

156

YUJI YOSHINO

˜˜˜y1 u˜ q u˜f­f ˜˜˜y1 q xu˜2 . This shows the equalTherefore ˜t9 s f˜˜tf˜y1 q f­f ity of matrices:

ž

f˜

u ˜f˜

0

f˜

/ž

­˜ x

yt˜ y­˜

/ž

f˜y1

˜y1 u˜ yf y1

f˜

0

/ ž s

­˜9 x

˜ yt9 . y­˜9

/

It follows from this that the matrix Ž f0˜ u˜f˜f˜ . gives a chain homomorphism LS ª R Ž F . ª LS ª R Ž F X . that is, of course, an isomorphism. v

v

v

v

The successive use of Ž2.3. shows the following: LEMMA Ž3.7.. The L-complex LT ª R Ž F . is quasi-isomorphic to F ; hence in the deri¨ ed category Dqf g ŽT . we ha¨ e an isomorphism LT ª R Ž F . ( F . v

v

v

v

v

v

Remark Ž3.8.. In Ž3.4., each step of constructing the L-complex needs only the information about the Eisenbud operators associated with Ti ª Tiy1. But we must note that LT ª R Ž F . cannot be determined only by the Eisenbud operators ˜t Ž1., ˜t Ž2., . . . , ˜t Ž r . associated with T ª R. Indeed it requires more information than that. v

v

To make this more explicit, we consider the case r s 2, so R s TrŽ x 1 , x 2 .T. In this case we have ­˜2 s x 1˜t Ž1. q x 2 ˜t Ž2.. Since ­˜3 s x 1˜t Ž1.­˜q x 2 ˜t Ž2.­˜s x 1 ­˜˜t Ž1. q x 2 ­˜˜t Ž2., and since  x 1 , x 2 4 is a regular sequence on S, there is a homomorphism ˜ s of degree y3 on F˜ such that v

˜t Ž1.­˜y ­˜˜t Ž1. s yx 2 ˜s,

˜t Ž2.­˜y ­˜˜t Ž2. s x 1 ˜s.

Considering the ring homomorphism Trx 1T ª R, we see that ­˜mT Trx 1T is a lifting map of ­ to Trx 1T and the Eisenbud operator of this is ˜t Ž2. mT Trx 1T. Thus the L-complex LT r x 1T ª R Ž F . has the differential map ˜Ž2. . ˜Ž1. s Ž x­˜ yty˜Ž2.­˜ . is a lifting homomorphism DŽ1. s Ž x­˜2 yt y ­˜ mT Trx 1T. Thus D 2 v

v

of DŽ1. to T, and since

2

˜Ž1. . s ŽD



­˜2 y x 2 ˜t Ž2. 0

˜t Ž2.­˜y ­˜˜t Ž2. ˜t Ž1. s x1 0 ­˜2 y x 2 ˜t Ž2.

0

ž

˜s , ˜t Ž1.

/

the Eisenbud operator of the complex LT r x 1T ª R Ž F . relative to T ª Trx 1T ˜s .; therefore the differential map of the is given by the matrix Ž ˜t Ž1. 0 ˜ t Ž1. v

v

L-COMPLEXES

157

AND WEAK LIFTINGS

successive L-complex LT ª T r x 1T LT r x 1T ª R Ž F . is the following: v

Fs

v



v

­˜ x2

yt˜Ž2. y­˜

yt˜Ž1. 0

ys˜ yt˜Ž1.

x1

0

y­˜

0

x1

yx 2

˜t Ž2. ­˜

0

.

Note that we need the homomorphism ˜ s in this construction, and that

˜s s 0 only when ˜t Ž1. commutes with ­˜.

Completely in a similar manner as this, we see that LT r x 2 T ª R Ž F . is ˜Ž1. . given by the differential map DŽ2. s Ž x­˜1 yt y ­˜ mT Trx 2 T, and hence the complex LT ª T r x 2 T LT r x 2 T ª R Ž F . is given by v

v

v

v

v

Cs



­˜ x1

yt˜Ž1. y­˜

yt˜Ž2. 0

˜s yt˜Ž2.

x2

0

y­˜

0

x2

yx 1

˜t Ž1. ­˜

0

.

Note that there is an equality of matrices 1



1 1 1

0

1

C

y1

1

y1

0

s F,

which shows that there is a chain isomorphism: LT ª T r x 1T LT r x 1T ª R Ž F . ( LT ª T r x 2 T LT r x 2 T ª R Ž F . . v

v

v

v

v

v

By virtue of Ž3.6., this shows that for any permutation  i1 , i 2 , . . . , i r 4 of  1, 2, . . . , r 4 , the L-complex successively taken from the sequence T ª Trx i1T ª Tr Ž x i1 , x i 2 . T ª ??? ª Tr Ž x i1 , x i 2 , . . . , x i r . T s R is chain isomorphic to that given in Ž3.4.. More generally we can prove the following: THEOREM Ž3.9.. For a free complex F in Kqf g Ž R ., the L-complex L F . is, up to chain isomorphism, uniquely determined only by an ideal I s xT, and independent of the choice of regular sequence that generates I. T ª RŽ v

v

v

158

YUJI YOSHINO

Proof. Let x9 s xX1 , xX2 , . . . , xXr be any other regular sequence that generates I. Setting the maximal ideal of T as m, note that a set of r elements of I generates I if and only if they give a base of the vector space Irm I and that any of those r elements is a regular sequence on T. It follows from this that there is a permutation  i1 , i 2 , . . . , i r 4 of  1, 2, . . . , r 4 such that the sets  x 1 , x 2 , . . . , x jy1 , xXi j , xXi jq 1, . . . , xXi r 4 Ž1 F j F r . form regular sequences which generate I. As we have shown above, the L-complex is independent of the order of the sequence; hence we can reduce the proof of the theorem to the case xX1 s x 1 , xX2 s x 2 , . . . , xXry1 s x ry1. In this case, let S s TrŽ x 1 , x 2 , . . . , x ry1 .T and we see that x r S s xXr S. By virtue of Ž3.6. it is sufficient to show that two L-complexes defined through x r and xXr are chain isomorphic to each other. If we write x for the image in S of an element x g T, then there is a unit u in S with xXr s ux r . Let ­˜ be a lifting map of ­ to S. Then the Eisenbud operators ˜t and ˜t9 are given by ˜ Therefore the following equality ­˜2 s x r ˜t s xXr ˜t9; hence we have ˜t s ut9. holds:

ž

­˜ xXr

˜ yt9 1 s ˜ 0 y­

/

0 u

ž

/ž

­˜ xr

yt˜ y­˜

/ž

1 0

0 . uy1

/

This shows the desired chain isomorphism is given by the matrix Ž 10 u0 .. PROPOSITION Ž3.10.. Let F be a free complex in Kqf g Ž R . as abo¨ e. Suppose that e FŽ i. ŽT ª R . s 0 Ž1 F i F r . in Ext 2R Ž F , F .. Then we ha¨ e a chain isomorphism of free complexes o¨ er R: v

v

LT ª R Ž F . mT R ( v

v

r

[ F wyi x v

v

r i

ž /.

is0

In this case, if F is a minimal complex o¨ er R, then LT ª R Ž F . is also minimal o¨ er T. v

v

v

Proof. The second assertion follows easily from the first. We prove the isomorphism in the proposition by induction on r. We have already shown the case r s 1 in Ž2.7.; hence assume that r G 2, and consider the rings:

6

R s TrŽ x 1 , x 2 , . . . , x r .T.

6

6

T1 s Trx 1T

T 1 s TrŽ x 2 , x 3 , . . . , x r .T

6

T

Before proceeding to the proof we make two remarks:

L-COMPLEXES

159

AND WEAK LIFTINGS

Ž3.10.1. Let F and F X be two complexes in Kqf g Ž R .. Since the functor Ext commutes with making direct sum, the elements e FŽ i. ŽT ª R . Žresp. e FŽ i.9 ŽT ª R .. Ž1 F i F r . in Ext 2R Ž F , F . Žresp. Ext 2R Ž F X , F X .. can be thought of as elements in Ext 2R Ž F [ F X , F [ F X .. Then we have an equality v

v

v

v

v

v

v

v

v

v

Ž i. Ž i. Ž i. e F[F 9 Ž T ª R . s eF Ž T ª R . q eF 9 Ž T ª R . ,

for any 1 F i F r. The proof of Ž3.10.1. is straightforward from the definition of the Eisenbud classes. Ž3.10.2. We have the following equality: e FŽ i. Ž T ª R . s

½

e FŽ1. Ž T 1 ª R .

for i s 1,

Ž T1 ª R .

for i G 2.

e FŽ i.

To show this, let ­˜ be a lifting map of ­ to T and let ­˜2 s Ý ris1 x i˜t Ž i.. Then ­˜mT T 1 is a lifting map of ­ to T 1 and Ž ­˜mT T 1 . 2 s x 1Ž˜t Ž1. mT T 1 .. Since e FŽ1. ŽT 1 ª R . is the homotopy equivalence class of Ž˜t Ž1. mT T 1 . mT 1 R, it equals e FŽ1. ŽT ª R .. The second equality in Ž3.10.2. can be proved similarly. We now return to the proof of the proposition. For brevity’s sake, we ˜ be a lifting map of D write Ž L , D . for the complex LT1 ª R Ž F ., and let D ˜2 s x 1˜t. By Ž3.10.2. we can apply the induction hypothesis to to T and let D T1 ª R to get the isomorphism v

v

v

ry1

L mT1 R (

Ž 3.10.3.

v

[ is0

ry1 i

ž /. F w yi x v

It then follows from Ž3.10.1. and Ž3.10.2. that the Eisenbud class of Ž r yi 1 . e FŽ1. ŽT ª R ., which is L mT1 R relative to T 1 ª R is equal to Ý ry1 is0 ˜ mT R s D mT1 R, we trivial by the assumption. On the other hand, since D ˜ mT T 1 is a lifting map of the differential map of L mT1 R to see that D ˜ mT T 1 . 2 s x 1Ž˜t mT T 1 .. Hence the Eisenbud class of L T 1, and that Ž D mT1 R relative to T 1 ª R is the homotopy equivalence class of ˜t mT R. Therefore we conclude from the above that the chain homomorphism ˜t mT R is homotopy equivalent to the zero map. Note that the differential map of LT ª R Ž F . s LT ª T1 Ž L . is, by definition, given by v

v

v

v

v

v

v

ž

˜ D x1

yt˜ . ˜ yD

/

160

YUJI YOSHINO

Thus LT ª R Ž F . mT R is obtained by the differential map v

v

ž

D mT1 R yt˜ mT R s ˜ 0 yD

˜ D x1

yt˜mT R

ž

/

yD mT1 R

/

;

therefore the complex LT ª R Ž F . mT R is a mapping cone of ˜t mT R: L mT1 R ª L mT1 Rwy2x. Since we have shown that ˜t mT R is homotopy equivalent to the zero map, it follows that LT ª R Ž F . mT R is a direct sum of L mT1 R and L mT1 Rwy1x; therefore the proposition is obtained by Ž3.10.3.. v

v

v

v

v

v

v

v

Before stating a main result in this section, we remark that the higher syzygy functors V n Ž n G 1. are defined inductively by V 1 s V and V nq1 s V V n. Note that for a minimal free complex G , V n G s t iqnG w n x if i s iŽ G .. v

v

v

v

THEOREM Ž3.11.. For a free complex F in Kqf g Ž R ., the following conditions Ža. ] Žd. are equi¨ alent. v

Ža. Žb. Žc. Žd.

F is weakly liftable to T. e FŽ i. ŽT ª R . s 0 in Ext 2R Ž F , F . for any 1 F i F r. F is liftable to TrI 2 . There is a chain isomorphism: v

v

v

v

TªR

L

v

r

Ž F . mT R ( [ v

is0

r i

ž/ F w yi x . v

If F is a minimal complex, then the abo¨ e conditions are equi¨ alent to the following condition: Že. There is a chain isomorphism: v

V r LT ª R Ž F . mT R ( v

v

r

r

[ V iF ž i / . v

is0

Proof. The equivalence of Žb. and Žc. is shown in Ž3.1.. The implication Žb. « Žd. has been proved in Ž3.10.. The proof of Žd. « Ža. and Že. « Ža. is immediate from the definition of weak liftings. Ža. « Žb.: The proof goes through in a manner similar to the proof of Ž2.7.Ža. « Žb.. Suppose there is a free complex G in Kqf g ŽT . that satisfies G mT R ( F [ F X for some complex Ž F X , ­ 9.. By this isomorphism each Gn is a lifting free module of Fn [ FnX and ­ G mT R s Ž ­0 ­09 .. Thus we can describe ­ G as v

v

v

v

v

ž

­˜

Ý ris1 x i u ˜Ž i.

Ý ris1 x i ¨˜Ž i.

­˜9

/

L-COMPLEXES

161

AND WEAK LIFTINGS

for some lifting maps ­˜ and ­˜9 of ­ and ­ 9, and for some homomorphisms u ˜Ž i., ¨˜Ž i. Ž1 F i F r . of degree y1. Let ­˜2 s Ýris0 x i˜t Ž i. and ­˜92 s Ý ris0 x i˜t9Ž i. as before. Then we have that G 2

0s Ž ­ . s

s

ž



­˜

Ý ris1 x i u ˜Ž i.

Ý ris1 x i ¨˜Ž i.

­˜9

/

­˜2 q Ý i , j x i x j u ˜Ž i. ¨˜Ž j.

Ý i x i Ž ­˜u ˜Ž i. q u˜Ž i.­˜9 .

Ý i x i Ž ¨˜Ž i.­˜q ­˜9¨˜Ž i. .

­˜9 2 q Ý i , j x i x j ¨˜Ž i. u ˜Ž j.

r

s

2

Ý xi is1



˜t Ž i. q Ý j x j u˜Ž i. ¨˜Ž j.

­˜u ˜Ž i. q u˜Ž i.­˜9

­˜9¨˜Ž i. q ¨˜Ž i.­˜

˜t9Ž i. q Ý j x j ¨˜Ž i. u˜Ž j.

0

0 .

Since  x 1 , x 2 , . . . , x r 4 is a regular sequence on T, this equality shows particularly that Ž˜t Ž i. q Ý j x j u ˜Ž i. ¨˜Ž j. . mT R s 0; hence t Ž i. s ˜t Ž i. mT R s 0 for any i. This implies condition Žb.. It remains to prove that the first four conditions imply Že. under the assumption that F is minimal. To show this we remark first that if G is any minimal free complex in Kqf g ŽT ., then VG mT R ( V Ž G mT R .; that is, the syzygy functor commutes with tensor. This is clear from the fact that iŽ G . s iŽ G mT R . for a minimal free complex G g Kqf g Ž R . and from the definition of syzygies. See Ž2.5.. Note also that LT ª R Ž F . is a minimal complex by Ž3.10.. Let i s Ž i LT ª R Ž F ... And now apply r times of V for both sides of the isomorphism in Žd. to get v

v

v

v

v

v

v

v

v

v

v

V r LT ª R Ž F . mT R ( V r Ž LT ª R Ž F . mT R . v

v

v

( t iqr

ž

r

(

v

v

js0

[ t iqr js0

r j

ž/ [ F wyj x r

r j

v

[ Ž t iqryj js0 r

(

r j

ž/ F .w r y jx v

r

[ V ryj F ž j / . v

js0

This completes the proof of the theorem.

wrx

ž/ Ž F wyj x . w r x

r

(

/

162

YUJI YOSHINO

Remark Ž3.12.. This theorem Ž3.11. sharpens the results of w2, Proposition Ž3.6.x. In fact, it follows from Ž3.11. that if an R-module M is weakly liftable to T, then there is an isomorphism of R-modules: r

V Tr

Ž M . mT R ( [ is0

V iR

r i

ž/ Ž M. .

See also Ž2.7..

4. MARANDA THEOREM FOR COMPLEXES As in Section 1 we write S for a Noetherian local ring and R s SrxS where x is a non-zero divisor on S. In this section, for given free complexes in Kqf g Ž S ., we would like to discuss when the non-isomorphism or the indecomposability of the complexes may be preserved by tensoring R over S. Note that such arguments about modules are called the Maranda theorem. To do this for complexes, we begin with the following lemma inspired by a result in w5, Proposition Ž1.1.x. LEMMA Ž4.1.. For a minimal free complex G in Kqf g Ž S ., the following three conditions are equi¨ alent. v

Ža. There is a chain isomorphism V LS ª R Ž G mS R . ( VG [ G . Žb. If iŽ H . G iŽ G . for a minimal free complex H in Kqf g Ž S ., then x Ext 1S Ž G , H . s 0. Žc. x Ext 1S Ž G , VG . s 0. v

v

v

v

v

v

v

v

v

v

v

Proof. For simplicity we write G for G mS R and L for LS ª R. Note first that G is a lifting of G to S, and thus G is liftable. In particular, the Eisenbud class of G is trivial; hence L Ž G . is a mapping cone of the multiplication map on G by x. v

v

v

v

v

v

v

v

v

v

v

Ža. « Žb.: Let H be a minimal free complex over S such that iŽ H . G i, where i s iŽ G .. Note that H j s 0 for j - i, since H is a minimal complex. Since G is a direct summand of V L Ž G ., Ext 1S Ž G , H . is also a direct summand of Ext 1S Ž V L Ž G ., H .. Thus we have only to show that x Ext 1S Ž V L Ž G ., H . s 0. Recall that any element of Ext 1S Ž V L Ž G ., H . is a homotopy equivalence class of a chain homomorphism f : V L Ž G . ª H of degree y1. We want to show that x f is homotopy equivalent to the zero map. Since G is a lifting complex of G , it follows from the definition that V L Ž G . and the chain homomorphism v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

L-COMPLEXES

163

AND WEAK LIFTINGS

f can be described in the following form: Giq1 [ Giq2

/

Gi [ Giq1

f iq 1 s Ž a iq 1 b iq 2 .

Hiy1 s 0

6

­ iH

??? ,

6

6

H ­ iq 1

0

f is Ž a i b iq 1 .

6

Hi

6

Hiq 1

G ­ iq 0 1 G x y ­ iq 2

6

f iq 2 s Ž a iq 2 b iq 3 .

???

ž

6

/

6

??? Giq 2 [ Giq3

G ­ iq 0 2 G x y ­ iq 3

6

ž

where a and b are homomorphisms G ª H wy1x and G ª H wy2x, respectively. Note that a n s bnq1 s 0 for n F i. By the commutativity of G . s ­nH Ž a nq1 , bnq2 . the above diagram we see that Ž a n , bnq1 .Ž ­ nqx 1 y ­0nq G 2 v

v

v

v

G for any n G i. In particular, we have x bnq 1 s ya n ­nq1 q ­nHa nq1 for n G i. Note from the above that this equality holds true even for n - i. ŽNotice here that the condition iŽ H . G iŽ G . is necessary to get x bi s H H ya iy1 ­ iG q ­ iy1 a i or equivalently ­ iy1 a i s 0.. Thus we have an equality of matrices v

Ž 0, a n .

ž

­nG

0

x

G y­nq 1

/

v

q ­nH Ž 0, a nq1 . s x Ž a n , bnq1 . ,

for any n. This shows that the map Ž0, a . gives a chain homotopy between x f and the zero map. Žb. « Žc.: This is obvious because iŽ VG . s iŽ G .. Žc. « Ža.: Consider the following chain homomorphism c : G ª VG wy1x: v

v

v

v

0

0

0,

6

6

Giq1

6

6

G ­ iq2

0

6

1

6

6

Giq2

Gi

6

1

???

G ­ iq1

Giq1

6

G ­ iq2

6

Giq2

6

???

where i s iŽ G .. It follows from condition Žc. that we have a homotopy G G map h such that x cnq 1 s h n ­nq1 q ­nq2 h nq1 for any n G i. Then there is a commutative diagram: v

Gi [ Giq1

0

ž h1 y10 /

/

i

6

ž

G ­ iq1

0

0

G y ­ iq2

/

Gi [ Giq1

0.

6

6

Giq1 [ Giq2

6

???

0 y1

/

6

iq1

G y ­ iq2

6

ž h1

0

x

6

Giq1 [ Giq2

6

???

ž

G ­ iq1

164

YUJI YOSHINO

This shows the chain homomorphism Ž h1 V L Ž G . ( G [ VG . v

v

v

0 y1

. gives an isomorphism

v

DEFINITION Ž4.2.. For a non-zero divisor x g S and for a given free complex G g Kqf g Ž S ., we define the integer n x Ž G . ŽF `. by v

v

n x Ž G . s inf  n N x n Ext 1S Ž G , VG . s 0 4 . v

v

v

Compare with w5, Sect. 1x. Note from Ž4.1. that if n G n x Ž G ., then we have x n Ext 1S Ž G , H . s 0 for any minimal free complex H with iŽ H . G iŽ G .. Now we can prove the following theorem that is the first part of the Maranda type theorem for complexes. v

v

v

v

v

v

THEOREM Ž4.3.. Let x g S be a non-zero di¨ isor on S and let G and GX be minimal free complexes in Kqf g Ž S .. Suppose that n is an integer with n G n x Ž G ., and also suppose that there is a chain isomorphism G mS Srx nq 1 S ( GX mS Srx nq 1 S. Then there is also a chain isomorphism G ( GX . v

v

v

v v

v

v

Proof. Remark that the consequence of the theorem says that the free complex F s G mS Srx nq 1S g Kqf g Ž Srx nq 1 S . has the unique lifting to S. Now assume that the complexes Ž G , ­˜. and Ž GX , ­˜9. in Kqf g Ž S . are as in the theorem. First note that iŽ G . s iŽ GX ., since both of them are minimal free complexes. Second, since they are the lifting complexes of F , we find a homomorphism u: ˜ G ª GX of degree y1 such that ­˜y ­˜9 s x nq 1 u. ˜ Thus we have v

v

v

v

v

v

v

v

v

˜˜9 s y­­ ˜˜9 x nq 1­˜u ˜ s ­˜2 y ­­ ˜˜9 y ­˜92 . s yx nq1 u˜­˜9. s y Ž ­­ Because x is a non-zero divisor on S, it follows that ­˜u ˜ s yu˜­˜9. Therefore we see that u ˜s is a chain homomorphism; hence its class is in Ext 1S Ž G , GX .. ŽSee Ž1.1... Now note from the definition of n x Ž G . and from Ž4.1. that x n annihilates the module Ext 1S Ž G , GX .. Thus x n u ˜s is homotopy equivalent to the zero map, i.e., there is a homomorphism ¨˜ of degree 0 with x n u ˜s s ­˜9¨˜ q ¨˜­˜. It then follows that v

v

v

v

v

­˜y ­˜9 s x nq 1 u ˜ s x Ž ­˜9¨˜s y ¨˜s­˜. . Therefore we have Ž1 q x¨˜s . ­˜s ­˜9Ž1 q x¨˜s ., which shows that the map 1 q x¨˜s is a chain homomorphism between Ž G , ­˜. and Ž G , ­˜9. and it is of course an isomorphism. v

v

L-COMPLEXES

165

AND WEAK LIFTINGS

The second part of the Maranda theorem for complexes is the next theorem. Before stating it, remark that the complex G is said to be indecomposable if there is no non-trivial direct decomposition of G . v

v

THEOREM Ž4.4.. Suppose that S is a complete local ring and that x is a non-zero di¨ isor on S as abo¨ e. Furthermore we assume that a free complex G in Kqf g Ž S . is indecomposable and n is an integer with n G n x Ž G .. Then the free complex G mS Srx nq 1 S o¨ er Srx nq 1 S is also indecomposable. v

v

v

Proof. Suppose G mS Srx nq 1S s G rx nq 1 G is decomposable. Then we can write v

v

­G s

ž

v

­Ž0.

x nq 1 uŽ0.

x nq1 ¨ Ž0.

X ­Ž0.

/

,

X where ­Ž0. and ­Ž0. are the differential maps of certain non-trivial free complexes, and uŽ0. and ¨ Ž0. are homomorphisms on G of degree y1. Compare with the proof of Ža. « Žb. of Ž2.7.. We prove the following claim by induction on l. v

Claim Ž4.4.1.. Ža l . There are homomorphisms fŽ1. , fŽ2. , . . . , fŽ l . on G of degree 0 such that fŽ i. g 1 q x i Hom S Ž G , G . for any i Ž1 F i F l .. ŽIn particular, they are invertible.. Žb l . There are homomorphisms on G of degree y1; ­Ž i. , ­ŽXi. , uŽ i. , and ¨ Ž i. for 0 F i F l with v

v

v

v

­ŽXi. ' ­ŽXiy1.

­Ž i. ' ­Ž iy1. ,

Ž mod x nq iq1 . ,

for any 1 F i F l. Žc l . If we set

­ŽGl . s Ž fŽ1. fŽ2. ??? fŽ l . .

y1

­ G Ž fŽ1. fŽ2. ??? fŽ l . . ,

then it has the following form:

­ŽGl . s

ž

­Ž l .

x nq lq1 uŽ l .

x nqlq1 ¨ Ž l .

­ŽXl .

/

.

Note from Žc l . that we have Ž G , ­ G . ( Ž G , ­ŽGl . .. As remarked in the above, the claim clearly holds for l s 0. Assume that claim Ž4.4.1. is true for an integer l. Then that Ž ­ŽGl . . 2 s 0 yields that v



v

­Ž2l . q x 2 nq2 lq2 uŽ l .¨ Ž l .

x nq lq1 Ž ­Ž l . uŽ l . q uŽ l . ­ŽXl . .

x nqlq1 Ž ­ŽXl .¨ Ž l . q ¨ Ž l . ­Ž l . .

­ŽXl2. q x 2 nq2 lq2 ¨ Ž l . uŽ l .

0

s 0.

166

YUJI YOSHINO

In particular we have

ž

uŽ l .

y¨ Ž l . s

s

ž ž

/ž

­Ž l .

x nq lq1 uŽ l .

x nqlq1 ¨ Ž l .

­ŽXl .

/

x nq lq1 uŽ l .¨ Ž l .

uŽ l . ­ŽXl .

y¨ Ž l . ­Ž l .

yx nq lq1 ¨ Ž l . uŽ l .

­Ž l . x

nqlq1

x nq lq1 uŽ l .

/ž

­ŽXl .

¨Ž l .

/ yuŽ l .

¨Ž l .

/

.

This shows that Ž y¨ Ž l . uŽ l . . s is a chain homomorphism on the complex Ž G , ­ŽGl . .. See Ž1.1.. Noting that Ž G , ­ŽGl . . ( Ž G , ­ G ., we see from the definition of n x Ž G . and from Ž4.1. that x n Ž y¨ Ž l . uŽ l . . s is homotopy equivalent to the zero map on Ž G , ­ŽGl . .. Hence there is a homomorphism Ž ag bd . v

v

v

v

v

on G such that the following equality holds: v

xn

ž

y¨ Ž l .

uŽ l .

/

s

ž

­Ž l . x

nqlq1

a y g

ž

x nq lq1 uŽ l .

/ž

a g

b d

/

¨Ž l .

­ŽXl .

b d

­Ž l .

x nq lq1 uŽ l .

x nqlq1 ¨ Ž l .

­ŽXl .

/ž

/

.

Particularly we see that x n uŽ l . ' ­Ž l . b y b­ŽXl .

Ž mod x nq lq1 . ,

y x n ¨ Ž l . ' ­ŽXl . g y g­Ž l .

Ž mod x nq lq1 . ;

hence, x nq lq1 uŽ l . ' ­Ž l . Ž x lq1b . y Ž x lq1b . ­ŽXl . y x nq lq1 ¨ Ž l . ' ­ŽXl . Ž x lq1 g . y Ž x lq1 g . ­Ž l .

Ž mod x nq 2 lq2 . , Ž mod x nq 2 lq2 . .

L-COMPLEXES

167

AND WEAK LIFTINGS

Therefore we have 1

ž

yx lq1 g =

ž

'

1

yx lq1b 1

1

ž

/ž

x lq1b 1

1

­Ž l . ­ŽXl .

/ž

­Ž l .

x nq lq1 uŽ l .

x nqlq1 ¨ Ž l .

­ŽXl .

1

/ž

x

/

lq1

g

1

/

/

Ž mod x nqlq2 . .

Now putting

fŽ lq1. s

ž

yx lq1b 1

1

s 1 q x lq1

ž

/ž

1 x

g

lq1

yb

g

/

/

1

q x 2 lq2

ž

ybg

/

,

we get from the above computation that there are homomorphisms ­Ž lq1. and ­ŽXlq1. of degree y1 on G such that v

G fy1 Ž lq1. ­ Ž l . fŽ lq1. s

ž

­Ž lq1.

x nq lq2 uŽ lq1.

x nqlq2 ¨ Ž lq1.

­ŽXlq1.

/

,

for some uŽ lq1. and ¨ Ž lq1. and that

­Ž lq1. ' ­Ž l . ,

­ŽXlq1. ' ­ŽXl .

Ž mod x nq lq1 . .

Therefore claim Ž4.4.1. is proved for l q 1. Now since S is a complete local ring, and since fŽ l . g 1 q x l Hom S Ž G , G ., the product fŽ1. fŽ2. ??? fŽ l . converges as l ª ` to a homomorphism F on G . Taking the limit of the equality in Žc l ., we have v

v

v

Fy1­ G F s

ž

­Ž`.

0

0

X ­Ž`.

/

,

X where ­Ž`. s lim l ª` ­Ž l . and ­Ž`. s lim l ª` ­ ŽXl . . This equality shows that the complex G is chain isomorphic to a certain non-trivial direct sum of two complexes. This is against the assumption; hence we deduce that G rx nq 1 G is indecomposable. v

v

v

168

YUJI YOSHINO

Remark Ž4.5.. Ža. Suppose that S is a complete local ring. Let us denote by C the full subcategory of Kqf g Ž S . that consists of all the free complexes of finite length. Then we can prove that C satisfies the Krull]Schmidt theorem, which claims that any complex in C is uniquely decomposed into a direct sum of indecomposable complexes. In order to show this, from the general argument, we have only to show that End S Ž G . is a local ring if G g C is indecomposable. A proof of this statement is outlined as follows: First we can show that End S Ž G . is module-finite over S for G g C. Suppose that End S Ž G . is non-local. Then there is f g End S Ž G . such that both of f and 1 y f are non-units in End S Ž G .. Let S be the image of the natural map S ª End S Ž G . and we denote by A the S-subalgebra of End S Ž G . generated by f . Note that A is a commutative S-algebra that is module-finite over S. But since A contains f and 1 y f , A is non-local. Then, since S is complete, A has a non-trivial idempotent. Therefore End S Ž G . also has a non-trivial idempotent, which means G decomposes non-trivially. Žb. There is a possibility of making an infinite direct sum within the category Kqf g Ž S .; for example, [ `is0 S wyi x is in Kqf g Ž S .. In general, we can say that any complex in Kqf g Ž S . decomposes into a direct sum of countably many indecomposable complexes. However, I do not know if the decomposition is unique or not. v

v

v

v

v

v

v

v

v

v

v

5. EISENBUD RESOLUTION FOR COMPLEXES Let S be a Noetherian local ring with maximal ideal m and let x g m be a non-zero divisor on S. We put R s SrxS as in Section 1. Now suppose we are given a free complex G in Kqf g Ž S .. If a free complex over R is quasi-isomorphic to G , then it is called an R-free resolution of G . ŽFor example, think of an S-free resolution of an R-module M as G , and then an R-free resolution of G is an R-free resolution of M in an ordinary sense.. In this section, we would like to construct an R-free resolution of G , which generalizes the Eisenbud resolution of an R-module. We begin with making the following v

v

v

v

v

v

DEFINITION Ž5.1.. ŽSee also Shamash w9x or Eisenbud w6x.. We say that a set S s  si N i G 04 of homomorphisms on G is a Shamash family of homomorphisms for G g Kqf g Ž S . with respect to x g S if the following four conditions are satisfied: v

v

Ža. For each i, si : G ª G is a homomorphism of degree 2 i y 1, Žb. s0 s ­ G , Žc. s0 s1 q s1 s0 s x, and Žd. Ý iqjsn si s j s 0 for n G 2. v

v

L-COMPLEXES

169

AND WEAK LIFTINGS

A Shamsh family of homomorphisms S s  si 4 is called minimal if si mS Srm s 0. Remark Ž5.2.. Ža. Shamash w9x shows that an S-free resolution of an R-module has a Shamash family of homomorphisms. But note that nonacyclic complexes over S may have a Shamash family of homomorphisms. Žb. Assume that G g Kqf g Ž S . has a Shamash family of homomorphisms with respect to x. Then the homologies of G are annihilated by x, that is, they are R-modules. v

v

Proof. Let z g G be a cycle. Then s0 Ž z . s ­ G Ž z . s 0; hence x z s Ž s0 s1 q s1 s0 . z s ­ G Ž s1Ž z .. is a boundary. v

We now consider the condition for a complex possessing a Shamash family of homomorphisms. Before we state the result, we must remark, to compare with the case of modules, that the negative Ext module may not vanish for complexes. For example, let I s Ž a1 , a2 , . . . , a r . be an ideal of S and consider the Koszul complex K associated to the sequence Ž a1 , a2 , . . . , a r .. Then it is easily seen that Extyr Ž K , K . ( 0 : S I, which S may be a non-trivial module. v

v

Let G be a free complex in Kqf g Ž S . as abo¨ e. Suppose that

LEMMA Ž5.3.. x

Ext 0S

v

ŽG , G . s 0 v

v

Ext 2S i Ž G , G . s 0

and

v

v

v

for i F y1.

Then G has a Shamash family of homomorphisms with respect to x. v

Proof. We can prove this essentially in the same way as Shamash w9x, and we give only an outline of the proof. We put s0 s ­ G and, since the multiplication by x on G is homotopy equivalent to zero, we find a homomorphism s1 of degree 1 with s0 s1 q s1 s0 s x. We construct inductively the homomorphisms s0 , s1 , . . . , sn satisfying degŽ si . s 2 i y 1 and v

Ž 5.3.1.

si s j s 0

Ý

for 2 F k F n.

iqjsk

By condition Ž5.3.1. we can show that Ý nis1 si snq1yi commutes with s0 ; hence it is a chain homomorphism of degree 2 n on G . Since nŽ Exty2 G , G . s 0, we can find a homotopy map snq 1 of degree 2 n q 1 S with s0 snq1 q snq1 s0 s yÝ nis1 si snq1yi . Thus the induction goes through. v

v

v

LEMMA Ž5.4..

Assume that G g Kqf g Ž S . satisfies the conditions: v

x g m Ann S Ext 0S Ž G , G . v

v

and

Ext 2S i Ž G , G . s 0 v

v

for i F y1.

Then G has a minimal Shamash family of homomorphisms with respect to x. v

170

YUJI YOSHINO

Proof. We also outline the proof. First we write x s Ý rjs1 a j x j , where a j g m and x j Ext 0S Ž G , G . s 0. By Ž5.3., we can construct a Shamash family of homomorphisms  siŽ j. N i G 04 with respect to each x j . Put s0 s ­ G and s1 s Ý rjs1 a j s1Ž j.. One can then show that for j - k, s1Ž j. s1Ž k . q s1Ž k . s1Ž j. commutes with s0 ; hence it is a chain homomorphism on G of degree 2. Ž G , G . s 0, we have a homotopy map SŽ j, k . with Since Exty2 S v

v

v

v

v

SŽ j, k . s0 q s0 SŽ j, k . s y Ž s1Ž j. s1Ž k . q s1Ž k . s1Ž j. . . Thus, putting s2 s Ý a2j s2Ž j. q Ý j- k a j a k SŽ j, k . , we see that Ý iqjs2 si s j s 0. To construct snq 1 for n G 2, assume that we have already got si Ž0 F i F n. with

Ž 5.4.1.

sk s

Ý

aJ SJ

and

JgNk

si s j s 0

Ý

for 2 F k F n,

iqjsk

where Nk s  J s Ž j1 , j2 , . . . , jk . N 1 F j1 F j2 F ??? F jk F r 4 , a J s a j1 a j 2 ??? a jk, and S J is a homomorphism of degree 2 k y 1 for J g Nk . Further we assume that for J g Nk Ž1 F k F n., s0 S J q S J s0 s y

Ž 5.4.2.

Ý Ž J 1 , J 2 .gK J

S J1 S J 2 ,

where K J s Ž J1 , J 2 . N J s J1 " J 2 , J1 / B, J 2 / B4 . It then follows from Ž5.4.1. that n

Ý snq 1yi si s Ý is1

aJ

JgNnq1

ž

Ý Ž J 1 , J 2 .gK J

/

S J1 S J 2 .

One can show from Ž5.4.2. that if J g Nnq 1 , then ÝŽ J1 , J 2 .g K J S J1 S J 2 commutes with s0 ; hence it is a chain homomorphism of degree 2 n. Thus by the assumption we can find a homotopy map S J with s0 S J q S J s0 s y

Ý Ž J 1 , J 2 .gK J

S J1 S J 2 ,

for J g Nnq 1. Then snq1 s Ý J g N nq 1 a J S J is a required homomorphism. Now in the following, we assume that the complex G in Kqf g Ž S . has a Shamash family of homomorphisms S s  si N i G 04 . We denote by Sw t x Žresp. Rw t x. the polynomial ring with the variable t, where we attach y2 as the degree of t and regard it as a graded ring. Similarly S²t : Žresp. R ²t :. denotes the divided power algebra having the variable t of degree 2. ŽRecall that S²t : s [ iG 0 St Ž i. as an S-module v

L-COMPLEXES

171

AND WEAK LIFTINGS

with t Ž0. s 1, t Ž1. s t , and t Ž i.t Ž j. s Ž i qi j .t Ž iqj... Note that Sw t x acts on S²t :, by which t sends t Ž i. to t Ž iy1. for i G 1 and sends 1 to 0. Of course the degree of the action of t is y2. Now we define the Eisenbud resolution: DEFINITION Ž5.5.. Let G be a free complex in Kqf g Ž S . that has a Shamash family of homomorphisms S s  si N i G 04 with respect x. v

Ža. We set

­ E˜ s

E˜ s S²t : mS G , v

v

Ý t i m si . iG0

˜ More precisely, E˜n s [ 2 iqjsn Ž St Ž i. mS Gj ., and ­ E Žt Ž i. m z j . s ˜ k Ž Ž i. . Ž i. Ýk G 0 t t m sk Ž z j . for t m z j g E˜n . We write E˜ Ž G , S . for Ž E˜ , ­ E .. Note that E˜ Ž G , S . may not be a complex. Actually, we have v

v

v

v

v

2

2

Ž ­ E˜ . s Ž 5.5.1.

ž

s

Ý t i m si iG0

/

Ý Ž t n m 1. Ý

ž

nG0

1 m si s j

iqjsn

/

s t m x s x Ž t m 1. . Žb. We define E s E Ž G , S . as E˜ Ž G , S . mS R, so that, v

v

E s R ²t : mS G v

v

v

v

ž i.e., E

n

­Es

s

v

[2 iqjsn Ž Rt Ž i. mS Gj . / ,

Ý t i m si . iG0

Note from Ž5.5.1. that E Ž G , S . is indeed a free complex over R. We say that E Ž G , S . is the Eisenbud resolution of G , because we shall show in Ž5.8. that it is quasi-isomorphic to G . v

v

v

v

v

v

Note that if G has a minimal Shamash family of homomorphisms S s  si 4 , then G must be a minimal complex. Indeed, since s0 s ­ G , we see that ­ G m Srm s 0. Note also that the Eisenbud resolution E Ž G , S . is a minimal free complex over R, if S is minimal. This is straightforward from the definition. v

v

v

v

172

YUJI YOSHINO

Remark Ž5.6.. We can define the natural map i: G ª E˜ Ž G , S . s ˜ S²t : mS G as z ¬ 1 m z . Note that we have i ­ G s ­ E i. In fact, v

v

v

v

­ E˜i Ž z . s ­ E˜ Ž 1 m z .

Ý Ž t k m sk . Ž 1 m z .

s

kG0

s 1 m s0 Ž z . s i­ G Ž z . . The natural projection S²t : ª R ²t : induces the homomorphism p : E˜ Ž G , S . ª E Ž G , S .. It follows from the above equality that the map j s p i : G ª E Ž G , S . is a chain homomorphism of complexes. v

v

v

v

LEMMA

v

v

v

Ž5.7.. There is an equality xE˜ Ž G , S . s Ž ­ E˜. 2 Ž E˜ Ž G , S ... v

v

v

E˜ . 2

E˜ . 2 Ž

v

Proof. Since Ž ­ s x Ž t m 1., we have Ž ­ E˜ Ž G , S .. : xE˜ Ž G , S .. To show the opposite inclusion, let z be any element in xE˜ Ž G , S .. We can write z s x ŽÝ k G 0 t Ž k . m z k . for some z k g G . Then consider j s Ý k G 0 t Ž kq1. m z k g E˜ Ž G , S . to get v

v

v

v

v

v

v

v

zsx

žÝt

Žk.

v

m zk

/

kG0

s x Ž t m 1.

žÝt

Ž kq1.

m zk

kG0

/

˜ 2

s Ž­ E. Ž j .. This shows the desired equality. Now we can show that the Eisenbud resolution E Ž G , S . actually resolves over R the complex G . v

v

v

THEOREM Ž5.8.. Suppose that a complex G g Kqf g Ž S . has a Shamash family of homomorphisms S s  si 4 . Then, under the same notation as in Ž5.6., the chain homomorphism j: G ª E Ž G , S . is a quasi-isomorphism. v

v

v

v

Proof. In this proof, for a complex G , we denote ZŽ G . s KerŽ ­ G . as the set of cycles, and B Ž G . s ImŽ ­ G . as the set of boundaries. And for the sake of convenience, write E , E˜ respectively, for E Ž G , S ., E˜ Ž G , S .. First we show that the map H Ž j .: H Ž G . ª H Ž E . is a monomorphism. To see this, let z g ZŽ G . and assume that jŽ z . g B Ž E .. We want to show that z g B Ž G .. It follows from the definition of j that iŽ z . g ­ E˜Ž E˜ . q xE˜ ; hence by Ž5.7. we have iŽ z . g ­ E˜Ž E˜ .. Thus there are v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

L-COMPLEXES

173

AND WEAK LIFTINGS

z k g G Ž k G 0. such that v

˜

iŽ z . s 1 m z s ­ E

žÝt

m zk .

Žk.

/

kG0

On the other hand, since ­ G Ž z . s 0, we see that ˜

0 s iŽ ­ G Ž z . . s ­ E Ž iŽ z . . ˜ 2

s Ž­ E.

ž Ý t mz / s x Ž t m 1. ž Ý t m z / sxž Ý t mz /. Žk.

k

kG0

Žk.

k

kG0

Ž ky1.

k

kG1

Because x is a non-zero divisor on S, we have z k s 0 for k G 1. Therefore ˜ iŽ z . s ­ E Ž1 m z 0 . s i ­ G Ž z 0 .. Noting that i is an injective map, we see that z s ­ G Ž z 0 . g B Ž G . as desired. Next we show that the map H Ž j . is an epimorphism. For this, let z be an arbitrary element in ZŽ E .. We must find w 0 g ZŽ G . that satisfies z ' jŽ w 0 . Žmod B Ž E ... We can write z as the image of p , that is, there is an element z˜s Ý k G 0 t Ž k . m z k in E˜ with p Ž z˜. s z . Then, since ­ E Ž z . s 0, we have ­ E˜Ž z˜. g xE˜ s Ž ­ E˜. 2 Ž E˜ ., and thus we can find j˜g E˜ with ­ E˜Ž z˜y ­ E˜Ž j˜.. s 0. In particular, we have v

v

v

v

v

v

v

v

v

˜

˜ 2

˜

x Ž t m 1 . z˜y ­ E Ž j˜. s Ž ­ E . z˜y ­ E Ž j˜. s 0;

ž

/

ž

/

˜ hence Ž t m 1.Ž z˜y ­ E Ž j˜.. s 0. Now write Ý k G 0 t Ž k . m w k Ž w k g G . for ˜ E z˜y ­ Ž j˜., and we obtain that Ý k G 1 t Ž ky1. m w k s 0, and therefore w k s 0 ˜ for k G 1. Consequently, we have z˜y ­ E Ž j˜. s 1 m w 0 s iŽ w 0 .. Here we ˜ must note that w 0 g ZŽ G ., since 1 m ­ G Ž w 0 . s i ­ G Ž w 0 . s ­ E iŽ w 0 . s E˜Ž ˜ E˜Ž ˜.. ­ z y ­ j s 0. Finally we see that v

v

z s p Ž z˜ . ˜

s p Ž ­ E Ž j˜. q i Ž w 0 . . s ­ E Ž p Ž j˜. . q j Ž w 0 . ' jŽ w0 . and this completes the proof.

Ž mod B Ž E . . , v

174

YUJI YOSHINO

Theorem Ž5.8. generalizes the theorem of Eisenbud w6, Theorem Ž7.2.x. In fact, if G is an S-free resolution of an R-module M, then E Ž G , S . gives an R-free resolution of M. v

v

v

Remark Ž5.9.. It is worth noting that the complex E Ž G , S . always has a natural chain endomorphism of degree y2. This endomorphism T : E Ž G , S . ª E Ž G , S .wy2x is defined by T Ž z . s Ž t m 1. z for any z g E Ž G , S .. Since Ž t m 1.ŽÝ k G 0 t k m sk . s ŽÝ k G 0 t k m sk .Ž t m 1., the map T is indeed a chain homomorphism. ˜ Note that ­ E is a lifting map of ­ E to S; hence it follows from Ž5.5.1. that T is the Eisenbud operator of the complex E Ž G , S .. Thus, by Ž2.7., E Ž G , S . is weakly liftable if and only if T is homotopy equivalent to the zero map. Furthermore, combining Theorem Ž5.8. with Ž2.3., we see that there is an isomorphism G ( LS ª R Ž E Ž G , S .. in the derived category Dqf g Ž S .. v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

LEMMA Ž5.10.. Under the same notation as in Ž5.9., the following is an exact sequence of complexes: im1

T

0 ª G mS R ª E Ž G , S . ª E Ž G , S . w y2 x ª 0. v

v

v

v

v

Proof. It is an easy exercise to show the injectivity of i m 1 and T Ž i m 1. s 0. To prove the lemma, let z s Ý k G 0 t Ž k . m z k g E Ž G , S ., where z k g Gk mS R. If T Ž z . s 0, then Ý k G 1 t Ž ky1. m z k s 0; hence z k s 0 for k G 1. Thus we have z s z 0 and this lies inside Ž i m 1.Ž G mS R ., which shows that the sequence is exact at the middle. Letting j s Ý k G 0 t Ž kq1. m z k , we see T Ž j . s z , and it follows that T is surjective. v

v

v

DEFINITION Ž5.11.. For a minimal free complex F , we define Hi nŽ F . by HiŽ F v .Ž F ., and call it the initial homology of F . Note that for any n G 0, Hi nŽ V n F . is the cokernel of ­ iŽFF v .qnq1. Hence, if F is a minimal R-free resolution of an R-module M, then Hi nŽ V n F . is the nth syzygy module of M. v

v

v

v

v

v

v

As a result of Lemma Ž5.10. we have the following COROLLARY Ž5.12.. If S is a minimal Shamash family of homomorphisms of G , then, for the complex E s E Ž G , S ., the Eisenbud operator T Ž defined in Ž5.9.. induces a sequence of surjecti¨ e maps of initial homologies of syzygies: v

v

??? ¸ Hi n Ž V nq 2 kq2 E

v

¸ Hi n Ž V nq 2 E

v

v

v

. ¸ Hi n Ž V nq 2 k E . ¸ ??? v

. ¸ Hi n Ž V n E . . v

L-COMPLEXES

AND WEAK LIFTINGS

175

Proof. It follows from Ž5.10. that T gives a surjective map Enq 2 ª En for any n; hence it maps the boundaries in degree n q 2 surjectively onto the boundaries of degree n. Hence T induces a surjective map CokerŽ ­ iŽEE .qnq3 . ª CokerŽ ­ iŽEE .qnq1 .. v

v

Remark Ž5.13.. Ding w4, Theorem Ž2.1.x has proven the following theorem: Let S be a Gorenstein local ring with maximal ideal m and let R s SrxS be as abo¨ e. For a finitely generated R-module M, if x belongs to m Ann S ŽEnd S Ž M .., then all the delta in¨ ariants d Ri Ž M . Ž i G 0. ¨ anish.

Note that d Ri Ž M . s 0 means exactly that V iR Ž M . is a surjective image of a stable maximal Cohen]Macaulay module over R. ŽWe say that a module is stable if it has no free direct summands.. The reader should refer to w10x for maximal Cohen]Macaulay modules and to w1x for delta invariants. Theorem Ž5.8. and Corollary Ž5.12. involve Ding’s theorem. In fact, if G is the minimal S-free resolution of an R-module M, then we have a minimal Shamash family of homomorphisms S by Ž5.4., since Ext Si Ž G , G . s Ext Si Ž M, M .. Thus, by Ž5.8., E Ž G , S . is a minimal R-free resolution of M. Then Ž5.12. claims that the Eisenbud operator of E Ž G , S . induces surjective maps: v

v

v

v

v

v

v

kq2 k i ??? ¸ V iq2 Ž M . ¸ V iq2 Ž M . ¸ ??? ¸ V iq2 R R R Ž M . ¸ VRŽ M .

for any i. Since it is clear that V nR Ž M . is a stable maximal Cohen]Macaulay module over R if n is larger than the dimension of R, we have d Ri Ž M . s 0 as desired. ŽCompare with Ding w4x.. Adding this remark, we give the following proposition that clarifies the meaning of having null delta invariants. PROPOSITION Ž5.14.. Let S be a regular local ring with maximal ideal m and let R s SrxS, where x is a non-zero di¨ isor of S. Ž Hence R is a ring of hypersurface.. Suppose that an R-module M has the following presentation o¨ er S, A

p

S n ª S m ª M ª 0, where A is the m = n matrix whose entries are in m. Ža. If d R0 Ž M . s 0, then there is an n = m matrix B ha¨ ing entries in m such that x Id S m s AB. ŽThe pair Ž A, B . is a matrix factorization of x in a weak sense.. Žb. Suppose that depth M G dim R y 1. Then the con¨ erse of Ža. holds.

176

YUJI YOSHINO

Proof. Note first that, since xM s 0, there is a homomorphism H: S m ª S n with x Id S m s AH. Putting N s ImŽ A. s V 1S Ž M ., we have an exact sequence 0 ª N ª S m ª M ª 0, which induces the exact sequence 0 ª Tor1S Ž M, R . ª N mS R ª R n ª M ª 0, where we must note that Tor1S Ž M, R . ( M. Therefore we have the exact sequence: c

0 ª M ª N mS R ª V1R Ž M . ª 0.

Ž 5.14.1.

The map c is defined as follows: Noting that any element of M is written as pŽ m . for some m g S m , c Ž pŽ m .. is the element AH Ž m . m 1 in N mS R. Second, we remark that N mS R has a finite projective dimension over R. In fact, if G is a finite S-free resolution of N, then G mS R is a finite R-free resolution of N mS R, since we can verify ToriS Ž N, R . s 0 for any i G 1. Now we suppose d R0 Ž M . s 0 to prove Ža.. Then, since N mS R has a finite projective dimension over R, a theorem of Auslander Žcf. Ding w4, Corollary Ž1.4.x. shows that the map c has its image in m Ž N mS R . s m NrxN. Thus, by the definition of c , we have AH Ž m . g m N s m ImŽ A. for any m g S m. Therefore, for any m g S m , we can find n g m S m with x m s AH Ž m . s An . In particular, fixing a basis  e1 , e2 , . . . , e m 4 of S m , we find n i g m S m Ž1 F i F m. such that xe i s An i . Setting the matrix B as Ž n 1 , n 2 , . . . , nm ., we have x Id S m s AB. To show Žb., let depth M G dim R y 1. In this case, V R Ž M . is a maximal Cohen]Macaulay module over R; hence the sequence Ž5.14.1. is the finite projective hull of M. ŽThe module in the middle is of finite projective dimension and the module on the right is maximal Cohen]Macaulay.. Note that we can follow the proof of Ža. the other way round, and thus if x Id S m s AB for some matrix B on m, we can show that ImŽ c . : m Ž N mS R .. Then a theorem of Auslander Žcf. Ding w4, Proposition Ž1.3.x. implies d R0 Ž M . s 0. v

v

Finally we remark that the following theorem of Ding w4, Theorem Ž2.7.x is easily obtained as a corollary of this proposition. COROLLARY Ž5.15. ŽDing.. Let S be a regular local ring and let R s SrxS be a ring of hypersurface as in Ž5.14.. For an R-module M generated by a single element, the following three conditions are equi¨ alent: Ža. Žb. Žc.

d R0 Ž M . s 0, d Ri Ž M . s 0 for any i G 0, x belongs to m Ann S Ž M ..

L-COMPLEXES

177

AND WEAK LIFTINGS

Proof. The implications Žc. « Žb. « Ža. are already shown in Ž5.13.. To show Ža. « Žc., we note that M has the following presentation: Ž a1 , a 2 , . . . , a n .

ª

Sn

S ª M ª 0.

Suppose d R0 Ž M . s 0. Then, by Ž5.14., there are bi g m Ž1 F i F n. with x s Ý nis1 a i bi . Since Ann S Ž M . s Ž a1 , a2 , . . . , a n . S, assertion Žc. holds. 6. COMPLEXITY AND VIRTUAL PROJECTIVE DIMENSION FOR COMPLEXES We introduce in this section the notion of complexity and virtual projective dimension for complexes that has been proposed for modules by Avramov w3x. And as an application of L-complexes we shall show that the Avramov’s equality also holds for a bounded complex. Let R be a local ring with residue field k as in the previous section. For any chain complex F over R, we denote iŽ F . s inf i N Hi Ž F . / 04 , and sŽ F . s sup s N Hs Ž F . / 04 . In this section we discuss everything in the derived category D f g Ž R . of the category of complexes with finitely generated homologies, and denote by D bf g Ž R . Žresp. Dqf g Ž R ., Dyf g Ž R .. the derived category of the category of bounded complexes Žresp. complexes bounded below, complexes bounded above.. See w8x for the general theory of derived categories. For a complex F g Dqf g Ž R ., we can define the Betti numbers of F according to Foxby w7x, v

v

v

v

v

v

v

BFR Ž n . s dim k TornR Ž F , k .

for any n g Z,

v

where TornR Ž F , k . denotes the nth homology of the complex F L mR k. ŽRecall that L m denotes the derived functor of m.. Furthermore the projective dimension of F is defined as follows: v

v

v

L

pd R F s s Ž F v

v

m k..

Similarly to the case of modules, the depth for a complex is defined Žsee Foxby w7x. depth R F s ys Ž RHom R Ž k, F . . v

v

for F g Dyf g Ž R . , v

where we should recall that RHom is a derived functor of Hom. Note that the depth is defined only for complexes that are bounded above, since otherwise the derived functor RHom would not be definite. Thus both the projective dimension and depth are simultaneously defined only for the complexes in D bf g Ž R .. Foxby w7, Proposition Ž3.13.x showed the following equality holds.

178

YUJI YOSHINO

LEMMA Ž6.1.. For a complex F g D bf g Ž R ., if pd R Ž F . - `, then pd R F s depth R y depth R F . v

v

v

v

DEFINITION Ž6.2.. ŽSee Avramov w3x.. Let F be a complex in Dqf g Ž R .. v

Ža. We say that F has complexity over R not bigger than c, and denote cx R Ž F . F c, if there is a real number e such that BFR Ž n. F en cy 1 for large n. Žb. We say that Ž S, x. is a deformation of R if x s  x 1 , x 2 , . . . , x r 4 : S is a regular sequence on S and R ( Srx S. And we define the ¨ irtual projecti¨ e dimension of F as vpd R F s inf pd S F N Ž S, x. is a deformation of R4 . v

v

v

v

v

For a finite R-module M, cx R Ž M . and vpd R M are understood as cx R Ž F . and vpd R F for a free resolution F of M. v

v

v

Remark Ž6.3.. For a complex F g easily verified. See Avramov w3, Lemma Ž3.4.x.

Dqf g Ž R .,

v

the following claims are

Ža. cx R Ž F . s 0 if and only if F has finite projective dimension. On the other hand, cx R Ž F . s 1 if and only if F has bounded Betti numbers. Žb. If Ž S, x. is a deformation of R, and if pd R Ž F . - `, then we have an equality pd S F s pd R F q r, where r is the length of the sequence x. Žc. If vpd R F s iŽ F ., then F is isomorphic in Dqf g Ž R . to mw R yiŽ F .x for some integer m. Žd. If pd R F - `, then vpd R F s pd R Ž F .. ŽThis is clear from Žb... v

v

v

v

v

v

v

v

v

v

v

v

v

v

Že. Let Ž S, x. be a deformation of R and assume that pd S F s vpd R F - `. Then the local rings R and S have the same embedding dimension, that is, x is contained in the square of the maximal ideal of S. Žf. The following four conditions are equivalent for R. v

v

Ži. vpd R G - ` for any G g D bf g Ž R .. Žii. vpd R M - ` for any finitely generated R-module M. Žiii. vpd R k - `. v

Živ.

v

R is a complete intersection.

Avramov w3, Theorem Ž3.5.x shows the following equality: LEMMA Ž6.4.. For a finitely generated R-module M, if vpd R M - ` then vpd R M s depth R y depth M q cx R Ž M .. We aim at proving the same equality for complexes in D bf g Ž R .. To do this, we first show the following

L-COMPLEXES

179

AND WEAK LIFTINGS

THEOREM Ž6.5.. Assume that R s SrxS, where S is a local ring and x is a non-zero di¨ isor of S. For F g Dqf g Ž R ., we ha¨ e the inequalities v

cx S Ž F . F cx R Ž F . F cx S Ž F . q 1. v

v

v

Furthermore if x is not contained in the square of the maximal ideal of S, then we ha¨ e cx S Ž F . s cx R Ž F .. v

v

The last assertion of the theorem follows from the following lemma. LEMMA Ž6.6.. Under the assumption of Ž6.5., if x is not contained in the square of the maximal ideal of S, then BFS Ž n . s BFR Ž n . q BFR Ž n y 1 . , for any n G 2. In particular cx S Ž F . s cx R Ž F .. v

v

Proof. In this case the residue field k of R is easily seen to be weakly liftable to S. See w2x. Hence we have V S Ž k . mS R s k [ V R Ž k .. Therefore, for any integer n G 2, L

TornS Ž k, F . s Hny 1 Ž F

ž

v

v

mR R . L mS V S Ž k .

/

R s Torny 1 Ž F , R mS V S Ž k . . v

s

R Torny 1

R Ž F , k . [ Torny 1Ž F , V R Ž k . . v

v

R R s Torny 1 Ž F , k . [ Torn Ž F , k . . v

v

B

The next lemma shows the first inequality in the theorem. LEMMA Ž6.7.. Under the circumstances of Ž6.5., we ha¨ e BFS Ž n . F BFR Ž n . q BFR Ž n y 1 . . Particularly, we see that cx S Ž F . F cx R Ž F .. v

v

Proof. We may assume that F is a minimal R-free complex. Let L s LS ª R Ž F . be the L-complex of F and let G be a minimal S-free resolution of F . Since L ( F in the derived category Dqf g Ž S ., we see that G is a direct summand of L . Hence we have v

v

v

v

v

v

v

v

v

v

v

BFS Ž n . s rank S Gn F rank S L n s rank R Fny1 q rank R Fn s BFR Ž n y 1 . q BFR Ž n . .

B

To complete the proof of the theorem, we must show the following lemma.

180

YUJI YOSHINO

LEMMA Ž6.8.. Under the same assumption as in Ž6.5., we ha¨ e BFR Ž n . F BFS Ž n . q BFS Ž n y 2 . q BFS Ž n y 4 . q BFS Ž n y 6 . q ??? . In particular, cx R Ž F . F cx S Ž F . q 1. v

v

Proof. We may assume that F is a minimal R-free complex, so that BFR Ž n. s rank R Fn for any n. Let L s LS ª R Ž F . be the L-complex of F . Note that L always has a Shamash family of homomorphisms S s  si N i G 04 which is given as follows: s0 s ­ L , s1 s Ž 00 10 . as a mapping L n s F˜ny1 [ F˜n ª L nq1 s F˜n [ F˜nq1 and si s 0 for i G 2. Indeed, it can be easily verified that the above defined S satisfies the conditions in Ž5.1.. Thus we may take the Eisenbud resolution E s E Ž L , S .. Note from the definition that En ( Fn [ Fny1 [ Fny2 [ Fny3 [ ??? . Note also from Ž2.3. and Ž5.8. that there are quasi-isomorphisms c : L ª F and j: L ª E . Define r : E ª F as a set  rn4 of natural projections of En onto Fn for any n. It is easy to see that r is a chain homomorphism and there is a commutative diagram: v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

F 6

6

c

L

v

r

j

6

E

E

v

v

Therefore we may conclude that r is also a quasi-isomorphism, and hence F ( E in Dqf g Ž R .. Now let G be a minimal S-free resolution of F . Then L contains G as a direct summand, that is, L ( G [ J where J is an S-free complex that is split exact, and the differential map ­ L is written as Ž ­0G ­0J .. Note J that, since J is split exact, we have rank Jn s rankŽ ­nq1 mS k . q rankŽ ­nJ . mS k . By the construction of the Eisenbud resolution we see that v

v

v

v

v

v

v

v

v

v

En s Ž L n [ L ny2 [ L ny4 [ ??? . mS R (  Ž Gn [ Jn . [ Ž Gny2 [ Jny2 . [ Ž Gny4 [ Jny4 . [ ??? 4 mS R and ­nE is given as a matrix

­L



s1

???

­

s1

???

­

s1 ???

L

L

??? ???

???

0

.

v

L-COMPLEXES

181

AND WEAK LIFTINGS

Therefore we have BFR Ž n . s Hn Ž E mR k . v

E s rank R En y  rank Ž ­nq1 mR k . q rank Ž ­nE mR k . 4

F

Ý Ž rank S Gny2 i q rank S

Jny2 i .

iG0 J J y Ý  rank Ž ­nq 1y2 i mR k . q rank Ž ­ny2 i mR k . 4

iG0

s

rank S Gny2 i q

Ý iG0

Jny2 i

Ý  rank S iG0

J J yrank Ž ­nq 1y2 i mR k . y rank Ž ­ny2 i mR k . 4

s

rank S Gny2 i s

Ý iG0

Ý BFS Ž n y 2 i . .

B

iG0

Now we prove Avramov’s equality for complexes as an application of Theorem Ž6.5.. THEOREM Ž6.9.. Let F be a complex in D bf g Ž R . and assume that vpd R F - `. Then the following equality holds: v

v

vpd R F s depth R y depth R F q cx R Ž F . . v

v

v

Proof. Let Ž S, x. be a deformation with pd S F s vpd R F - `. Then by Foxby’s equality Ž6.1. we have pd S F s depth S y depth S F . On the other hand, if r is the length of the sequence x, then it follows from successive use of Ž6.5. that cx R Ž F . F r. Noting that depth S s depth R q r and depth S F s depth R F , one obtains the inequality: v

v

v

v

v

v

v

vpd R F s pd S F G depth R y depth R F q cx R Ž F . . v

v

v

v

To show the other inequality, we may assume that F is a minimal complex of R-free modules. Let s be an integer which is large enough so that ts F is acyclic. ŽSuch an integer exists because F g D bf g Ž R ... Since F is an R-free complex, ts F is an R-free resolution of a certain R-module M. Note that the commutative diagram of R-free complexes v

v

v

v

v

FiŽ F v . 6

6

FiŽ F v .

0

6

0

6

6

???

6

Fsy1

???

6

Fsy1 6

6

0

0 6

6

6

F 6s

6

F 6s

6

Fsq 1

6

???

Fsq 1 6

6

???

182

YUJI YOSHINO

yields the short exact sequence Žor the triangle. of the complexes in D bf g Ž R .: 0 ª F X ª F ª M w ys x ª 0.

Ž 6.9.1.

v

v

X

Note that F is of finite projective dimension by its definition and that F and M have the same complexity, since F and ts F have the same nth Betti number if n G s. Note also that M has a finite virtual projective dimension. In fact, since vpd R F - `, there is a deformation Ž S, x. of R with pd S F - `; hence noting from Ž6.3.Žb. that pd S F X - `, we have pd S M - `. Now we take a deformation Ž S, x. of R so that vpd R M s pd S M - `. Then, since pd S F - `, we can apply Foxby’s equality Ž6.1. to get pd S F s depth S y depth S F s depth R q r y depth R F , where r is the length of x. On the other hand, by Avramov’s equality Ž6.4. for a module M, we have v

v

v

v

v

v

v

v

v

v

v

pd S M s vpd R M s depth R y depth M q cx R Ž M . . But note that pd S M s depth S y depth M; hence we obtain cx R Ž F . s cx R Ž M . s r. Thus we have pd S F s depth R q cx R Ž F . y depth R F . Since vpd S F F pd S F , we have the desired inequality. v

v

v

v

v

v

Remark Ž6.10.. Let M be a finitely generated R-module. We can define the virtual injective dimension vid R M of M the same way as in Ž6.2., vid R M s inf  id S M N Ž S, x . is a deformation of R 4 , where id denotes the injective dimension. See Avramov w3, Sect. 5x. Now we denote by M k the dual RHom R Ž M, D . of M, where D is the dualizing complex of R. When D is taken in the normalized position, we can easily see that v

v

v

Ž 6.10.1.

dim k TornR Ž M k , k . s dim k Ext Rn Ž M k , k . s dim k Ext Rnq dim R Ž k , M . ;

hence we have the equalities: pd R M ks id R M y dim R and depth M ks dim R. Note that this leads us to vid R M s vpd R M kq dim R. Note also from Ž6.10.1. that cx R Ž M k. represents the degree of the polynomial growth of the Bass numbers of M, which Avramov calls the prexity of M and denotes by px R Ž M .. Thus we can apply Theorem Ž6.9. to M k and we get the second equality of Avramov w3, Theorem Ž5.2.x: vid R M s depth R q px R Ž M . .

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AND WEAK LIFTINGS

183

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