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Some Remarks on Index and Generalized Loewy Length of a Gorenstein Local Ring
187, 150]162 Ž1997. JA976770
JOURNAL OF ALGEBRA ARTICLE NO.
Some Remarks on Index and Generalized Loewy Length of a Gorenstein Local Ring Mitsuyasu Hashimoto* and Akira Shida† Department of Mathematics, School of Science, Nagoya Uni¨ ersity, Chikusa-ku, Nagoya 464-01, Japan Communicated by Mel¨ in Hochster Received March 29, 1995
1. INTRODUCTION Throughout this paper, the word ‘‘ring’’ will mean commutative noetherian ring with 1. Let R be a Gorenstein local ring. M. Auslander introduced the notion of a d-invariant d R Ž M . for a finitely generated R-module M. It is defined as the smallest integer m such that there exists an epimorphism X [ R m ª M with X a maximal Cohen]Macaulay module which has no non-zero free summand. In studying the d-invariant, Ding w10]13x studied the notion of index. The index of R, denoted by indexŽ R ., is defined as the smallest positive integer such that d R Ž Rrm n . ) 0. He compared indexŽ R . with the generalized Loewy length l l Ž R . of R, which is defined as the minimum of the Loewy lengths of Rrx R for all systems of parameters x of R, and he conjectured Žand proved for some special cases. that indexŽ R . s l l Ž R . in general Žsee Definition 2.7 and Remark 3.1.. In this paper, we first remark that the generalized Loewy length is not stable even under a finite ´ etale extension, but stable under completion ŽSection 3.. We need at least to assume that the residue field of R is infinite to study Ding’s conjecture. Next, we prove that the minimal Cohen]Macaulay approximation, which is indispensable when we study d-invariants, of a finitely generated module M over a Cohen]Macaulay local ring R is preserved by an extension by * Current address: Nagoya University College of Medical Technology, 1-1-20 Daikominami, Higashi-ku, Nagoya 461, Japan. E-mail address: [email protected]. † E-mail address: [email protected]. 150 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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a Gorenstein local homomorphism R ª S under the assumption ToriR Ž S, M . s 0 Ž i ) 0. ŽProposition 4.3.. If R is Gorenstein moreover, then the higher delta invariants are also preserved ŽCorollary 4.6.. This unifies two known important cases: the flat case and the case RS s RrxR with x both R-regular and M-regular. We note that many of the notions and the results on flat morphisms have been generalized to the situation of finite flat dimension w4, 5, 3x. Let Ž R, m . and Ž S, n . be Gorenstein local rings, and w : R ª S be a local homomorphism of finite flat dimension Ži.e., the flat dimension of S as an R-module is finite.. Recently, the second author proved the following. THEOREM 1.1 w17, Ž3.7.x.
Let w : R ª S be as abo¨ e. Then we ha¨ e
index Ž R . F index Ž S . . This can be seen as a sort of flat descent Žgeneralized to the finite flat dimension context.. A good property ‘‘index F n’’ of S is inherited by R for any n. In Section 5, assuming S is R-flat, we prove indexŽ S . F indexŽ R . ? l l Ž F . Ž F [ Srm S ., which can be seen as a counterpart of the theorem above. This is our main theorem. Furthermore, this inequality shows that if the closed fiber F is a regular local ring, we have indexŽ R . s indexŽ S . as might be expected. We cannot expect that the fiber ring F s Srm S has a small index even if S has a small index. This can be seen by the following example: R s k ww x t xx ; k ww x xx s S, where t is an arbitrary positive integer. However, if S is artinian, then we have an inequality: l l Ž R . q l l Ž F . y 1 F l l Ž S . F l l Ž R . ? l l Ž F .. This will be proved in Section 6 for artinian rings which are not necessarily Gorenstein.
2. PRELIMINARIES Unless otherwise specified, we assume that Ž R, m , k . is a Cohen]Macaulay local ring with the canonical module K R . By Rˆ we mean the completion of R with respect to the maximal ideal m. For an ˆ denotes Rˆ mR M. For a proper ideal I of R, we denote R-module M, M the associated graded module [i G 0 I i MrI iq1 M by Gr I M. Let Ž S, n . be a local ring, and w : R ª S a homomorphism. We say that w is local when w Ž m . ; n. PROPOSITION 2.1. Let R be a Cohen]Macaulay local ring, X a maximal Cohen]Macaulay R-module, and Y Ž resp. Z . a finitely generated R-module of finite injecti¨ e Ž resp. projecti¨ e . dimension. Then we ha¨ e Ža. Ext iR Ž X, Y . s 0 for i ) 0; Žb. ToriR Ž Z, X . s 0 for i ) 0.
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Proof. Part Ža. is noted in w18x. Part Žb. is a slight generalization of w16, Ž2.6.x, and follows easily from w8x. Let M be a finitely generated R-module. A sequence of finitely generated R-modules f
g
0 ª Yª XªM ª 0
Ž 2.2.
is called a Cohen]Macaulay approximation if X is a maximal Cohen]Macaulay R-module, Y is of finite injective dimension, and the sequence is exact. The Cohen]Macaulay approximation Ž2.2. is said to be minimal when X and Y have no non-zero direct summand in common through f. It is said to be right minimal when g is right minimal, that is, for any non-isomorphic map w : X ª X, we have g ( w / g. The next lemma and its corollary seem to be well known, but we state them with proofs. The complete case Žwhich is essential . is found in w19x. LEMMA 2.3.
Let f
g
A [ 0 ª Yª XªM ª 0 be a sequence of finitely generated R-modules. Then, the following conditions are equi¨ alent. Žrmin. A is a right minimal Cohen]Macaulay approximation of M. Žmin. A is a minimal Cohen]Macaulay approximation of M. $
ˆ is a right minimal Cohen]Macaulay approximation of M. ˆ Žrmin. A $
ˆ is a minimal Cohen]Macaulay approximation of M. ˆ Žmin. A $
Proof. It is easy to see that A is a Cohen]Macaulay approximation if and only if A is a Cohen]Macaulay approximation. Thus, the problem is the minimality. Žrmin. « Žmin.. Assume that X s X 0 [ X 1 and 0 / X 0 ; Im f. We define w : X ª X by w Ž x 0 q x 1 . s x 1 for x 0 g X 0 and x 1 g X 1. Then, we have g ( w s$ g, and w is not isomorphic. Žmin. « Žmin.. See Proposition 4.3 below. $ $ ˆ s A. Assume that Žmin. « Žrmin.. We may assume Rˆ s R and A w g End R Ž X . is not an isomorphism, and g ( w s g. We have X / 0, and hence M / 0 by minimality of A. We set L to be the sub-R-algebra of End R Ž X . generated by R and w . Note that L is commutative, and is module finite over R. As w preserves the kernel of g, we may and shall regard Y and M as L-modules so that f and g are L-linear. Note that w g L acts as an identity map on M. By Nakayama’s lemma, w is not contained in the radical of L, since w Ž M . s M and M / 0. As w is not a
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unit in L Žsince w is not an isomorphism on X ., we have that L is not local. It follows that L has a non-trivial idempotent, say e / 0, 1, since R is henselian. With replacing e with 1 y e if necessary, we may assume that the image of e in LrŽ1 y w . L, which is local, is a unit. As e acts as an automorphism on M, it is easy to see that X 0 [ ImŽ1 y e . s Ker e is a non-zero direct summand of X which is contained in Im f. $ Žrmin. « Žrmin.. Assume that w g End R Ž X . is non-isomorphic and that g ( w s g. Then, w ˆ is non-isomorphic and ˆg ( wˆ s ˆg. COROLLARY 2.4. Let M be a finitely generated R-module. Then, there exists a minimal Cohen]Macaulay approximation of M, uniquely up to isomorphism. Proof. A Cohen]Macaulay approximation of M exists, see w1x. Removing non-zero common direct summands through f if any, we obtain a minimal one from this. The uniqueness of the right minimal Cohen]Macaulay approximation is shown by a standard comparison argument in w1x. For an R-module M, we define the f-rank of M, denoted by f-rank R M, to be the number max i < R i is a direct summand of M 4 . DEFINITION 2.5. Let R be a Cohen]Macaulay local ring with K R , and let 0 ª Y ª X ª M ª 0 be the minimal Cohen]Macaulay approximation of M over R. We define the d-in¨ ariant of M, denoted by d R Ž M ., as f-rank R X. And we define the index of a Gorenstein local ring R, denoted by indexŽ R ., as the integer min n < d R Ž Rrm n . / 04 . For a nonnegative integer n, the d-invariant of the nth syzygy V nR Ž M . of M is denoted by d Rn Ž M ., and we call it the nth d-in¨ ariant of M. PROPOSITION 2.6 w10x. modŽ R .. Then
Let R be a Gorenstein local ring and M Ž/ 0. g
Ža. If pd R M is finite, then d R Ž M . s m Ž M .Ž) 0., where m Ž M . is the minimal number of generators of M. Žb. If M ª N ª 0 is exact, then d R Ž M . G d R Ž N .. Next we define the generalized Loewy length. DEFINITION 2.7. Let M be an R-module of finite length. The Loewy length of M, denoted by l l R Ž M ., is the smallest integer n such that m n M s 0. We define the generalized Loewy length of R, which is also denoted by l l Ž R ., as the minimum of all integers l l R Ž RrŽx.., where x is a system of parameters of R.
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3. REMARKS ON GENERALIZED LOEWY LENGTH Remark 3.1. Ding proved that indexŽ R . F l l Ž R . for any Gorenstein local ring R w11x. He conjectured that indexŽ R . s l l Ž R . for the arbitrary Gorenstein local ring R. He claimed that the conjecture is true when the associated graded ring Gr m R of R is a Cohen]Macaulay ring w12x. He also claimed that the conjecture is true when R is gradable and depth Gr m R G dim R y 1 w13x. These are certainly true when Rrm is infinite. If we drop the condition on the residue field, there is a counterexample to the conjecture. EXAMPLE 3.2. Let k s F 2 , and K s F4 , where Fq is the q-element field. When we set S s k ww x, y xx, f s xy Ž x q y . g S, and R s SrŽ f ., then we have 4 s l l Ž R . ) l l Ž K mk R . s index Ž K mk R . s index Ž R . s 3. Proof. As the rings in consideration are hypersurfaces, we have that indices equal to multiplicities w10x, and we have indexŽ K mk R . s indexŽ R . s 3. Let v g K _ k, and set z s x y v y. Then, we have l l ŽŽ K mk R .rŽ z .. s 3. Hence, we have l l Ž K mk R . s 3. As we have l l Ž RrŽ x y y 2 .. s 4, we have l l Ž R . F 4. It remains to show that l l Ž R . ) 3. Assume the contrary. Then, there exists some R-regular element z g m such that m 3 ; Ž z ., where m s Ž x, y . R. We set r s max i < z g m i 4 . It is easy to see that r F 2. If r s 2, then zR is annihilated by m 2 , where R s Rrm 4 . Hence, we have 3 s l R Ž Rrm 2 . G l R Ž Rz . G l R Ž m 3 R . s l R Ž m 3rm 4 . s 3. This shows zR s m 3 R, and hence z g m 3. This contradicts r s 2. Consider the case r s 1. Take any preimage z g S of z. Letting an appropriate element in GL 2 Ž k . act on S, we may assume that z s x y g Ž x, y . with g g ŽŽ x, y . S . 2 without loss of generality. As RrŽ z . is a hypersurface, it suffices to show that l R Ž RrŽ z .. G 4 to lead to a contradiction. When we set R9 s SrŽ z ., then R9 is a discrete valuation ring, and RrŽ z . s R9rŽ f ., where f s xy Ž x q y . is the image of f in R9. On the other hand, we have y g Ž x, y . R9, x q y g Ž x, y . R9, and x s g g ŽŽ x, y . R9. 2 . Hence, we have f g ŽŽ x, y . R9. 4 and we have l Ž R9rŽ f .. G 4. As we have seen, the generalized Loewy length is not stable under a finite ´ etale extension. However, it is stable under completion.
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LEMMA 3.3. We ha¨ e l l Ž R . s l l Ž Rˆ.. Proof. If x is a system of parameters of R such that l l Ž R . s ˆ Rˆ. s l l R Ž Rrx R . s l l Ž R .. l l R Ž Rrx R ., then we have l l Ž Rˆ. F l l RˆŽ Rrx So it suffices to show l l Ž R . F l l Ž Rˆ.. Let n s l l Ž Rˆ., and take a system of parameters Ž ˆ x1, ˆ x2 , . . . , ˆ x d . ; Rˆ such that m n Rˆ ; Ž ˆ x1, ˆ x2 , . . . , ˆ x d ., where d s dim R. Then for each ˆ xi Ž i s 1, 2, . . . , d ., we can choose an element x i g R such that x i y ˆ xi g ˆ We claim that m n ; Ž x 1 , x 2 , . . . , x d .. In fact since m n Rˆ ; m nq 1 R. ˆ we have Žˆ x1, ˆ x2 , . . . , ˆ x d . ; Ž x 1 , x 2 , . . . , x d . Rˆ q m nq 1 R, m n Rˆ ;
ˆ F ž Ž x 1 , x 2 , . . . , x d . Rˆ q m nq i Rˆ/ s Ž x 1 , x 2 , . . . , x d . R. i)0
Hence, we have m n s m n Rˆ l R ; Ž x 1 , x 2 , . . . , x d . Rˆ l R s Ž x 1 , x 2 , . . . , x d . as desired.
4. A GORENSTEIN HOMOMORPHISM AND MINIMAL COHEN]MACAULAY APPROXIMATIONS Let w : R ª S be a flat local homomorphism of Gorenstein local rings. If 0 ª Y ª X ª M ª 0 is a Žminimal. Cohen]Macaulay approximation over R, then so is 0 ª S mR Y ª S mR X ª S mR M ª 0. In particular, the Žhigher. d-invariants are preserved by this base extension. In this section, we generalize this to the context of morphisms of finite flat dimension ŽProposition 4.3, Corollary 4.6.. The case S s RrxR with x both R- and M-regular w2, Ž5.1.; 20, Ž1.8.x, as well as the flat case w20, Ž1.5., Ž1.7.x, has been well known. First, we introduce the notion of a Gorenstein local homomorphism Žsee w4x.. Let w : Ž R, m . ª Ž S, n . be a local homomorphism. We say that w is a Gorenstein local homomorphism if w is of finite flat dimension Ži.e., fd R S - `., and m Ri s mSiqs for some s and arbitrary i G 0, where m Ri s dim k Ext Ri Ž k, R . is the ith Bass number. Next, we define a Cohen factorization which was introduced in w6x. DEFINITION 4.1. Let w : Ž R, m . ª Ž S, n . be a local homomorphism of local rings. We say w is factorizable if it can be decomposed as w s st with t : R ª T flat and TrmT a regular local ring, s : T ª S surjective, and T a complete local ring; often we shall refer to such a situation by saying that w s st is a Cohen factorization.
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It is known that if S is a complete local ring, then there exists a Cohen factorization Žsee w6, Ž1.1.x.. Let w : R ª S be a local homomorphism, and w s st its Cohen factorization. Then, w is of finite flat dimension if and only if so is s w6, Ž3.3.x. Note also that, if w is module finite, then we have fd R S - ` m ToriR Ž S, Rrm . s 0 for i 4 0 m pd R S - `. By w4, Ž4.2., Ž4.6.x, w is Gorenstein if and only if so is s . LEMMA 4.2. Let w : R ª S be a local homomorphism of Cohen]Macaulay local rings. We assume that fd R S - `. Let X be a maximal Cohen]Macaulay R-module. Then we ha¨ e ToriR Ž S, X . s 0 for i ) 0. We also ha¨ e S mR X is a maximal Cohen]Macaulay S-module. Proof. We may assume that S is complete. If S is R-flat, then the assertion is obvious. Consider the general case. Take a Cohen factorization w s st . As the lemma is true for t , we may replace w by s , and we may assume that w is surjective. Now, the vanishing ToriR Ž S, X . s Ž i ) 0. is Proposition 2.1Žb.. As for the last assertion, the same proof in w17, Ž3.1.x works. PROPOSITION 4.3. Let w : R ª S be a Gorenstein local homomorphism of Cohen]Macaulay local rings, and M a finitely generated R-module. Furthermore, we assume that R has the canonical module K R , and ToriR Ž S, M . s 0 f
for all i ) 0. If 0 ª Y ª X ª M ª 0 is the minimal Cohen]Macaulay approximation of M o¨ er R, then the sequence SmR f
S mR X ª S mR M ª 0
6
0 ª S mR Y
Ž 4.4.
is the minimal Cohen]Macaulay approximation of S mR M o¨ er S. Remark 4.5. When w is surjective and S s RrI, then, for r G 0, we have ToriR Ž S, M . s 0 Ž i ) r . if and only if depth R Ž I, M . G pd R S y r Ždepth sensitivity of perfect ideals, see w9x.. In particular, the Tor-independence assumption ToriR Ž S, M . s 0 Ž i ) 0. above is equivalent to depth R Ž I, M . s pd R S. Generalizing this to the imperfect case, the second author proved the following, which we only state the result here: Let R be a Žnot necessarily Cohen]Macaulay. ring, I an ideal of R of finite projective dimension, and S s RrI. If M is a finitely generated R-module with MrIM / 0, then we have sup i < ToriR Ž S, M . / 0 4 ssup pd R pSp ydepth R pMp < pg supp Ž MrIM . 4 . COROLLARY 4.6. In the proposition, assume moreo¨ er that R is Gorenstein. Then we ha¨ e f-rank S Ž S mR X . s f-rank R X . In particular, we ha¨ e
dSn Ž S
mR M . s
d Rn Ž M ..
Ž 4.7.
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Proof. The equality Ž4.7. follows from the proposition and w17, Ž3.1.x. As we have ToriR Ž S, M . s 0 Ž i ) 0. by assumption, we have V Sn Ž S mR M . ( S mR V nR Ž M ., and we have dSn Ž S mR M . s d Rn Ž M .. Proof of Proposition 4.3. Since it is sufficient to prove that ˆ Rf Sm
Sˆ mR X ª Sˆ mR M ª 0
6
0 ª Sˆ mR Y
is the minimal Cohen]Macaulay approximation of Sˆ mR M, we may assume that S is complete. We proceed in several steps. Step 0. Note that ToriR Ž S, K R . s 0 for i ) 0 by Lemma 4.2. We have that S mR K R ( K S , since w is Gorenstein w4, Ž5.1.x. Step 1. Since ToriR Ž S, M . s 0 for i ) 0 by assumption and RŽ Tori S, X . s 0 by Lemma 4.2, we have ToriR Ž S, Y . s 0 for i ) 0. Step 2. We show that S mR Y is of finite injective dimension over S. First note that a module Y is of finite injective dimension if and only if there exists an exact sequence wn
w ny1
w1
w0
0 ª In ª Iny1 ª ??? ª I0 ª Y ª 0
Ž 4.8.
such that I j is a finite direct sum of K R Ž j s 0, 1, . . . , n.. As the modules in Ž4.8. are Tor-independent of S by Step 0 and Step 1, the sequence ???
S mR Y
0
6
XmR w 0
SmR w ny1
6
S mR Iny1 6
S mR I0
6
SmR w 1
SmR w n
6
0 ª S mR In
is exact. Here S mR I j is a finite direct sum of K S by Step 0. Thus, we conclude that S mR Y is of finite injective dimension as an S-module. Step 3. We know from Step 2 that S mR Y is of finite injective dimension. Since S mR X is a maximal Cohen]Macaulay S-module by Lemma 4.2, and Ž4.4. is an exact sequence by Tor1R Ž S, M . s 0, we have that Ž4.4. is a Cohen]Macaulay approximation of S mR M over S. Step 4. Next, we show that the canonical homomorphism S mR Hom R Ž K R , Y . ª Hom S Ž S mR K R , S mR Y .
Ž 4.9.
given by a m f ¬ a ? Ž S mR f . for a g S and f g Hom R Ž K R , Y . is surjective. As this is obvious when S is R-flat, we may assume that w is surjective Žconsider the Cohen-factorization of w . so that S s RrI for a proper ideal I of R of finite projective dimension. Note that 0 ª I mR Y ª Y ª S mR Y ª 0 is exact since i ) 0.
Tor1R Ž S, Y
. s 0, and that
ToriR Ž I, Y
.(
Ž 4.10. R Ž Toriq1 S, Y
. s 0 for
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Let F be a finite free resolution of I. Then, as we have that F mR Y ª I mR Y is quasi-isomorphic and that each term of F mR Y is of finite injective dimension, we have that the injective dimension of I mR Y is finite. From the exact sequence Ž4.10., we obtain an exact sequence Hom R Ž K R , Y . ª Hom R Ž K R , S m Y . ª 0 s Ext 1R Ž K R , I mR Y . by Proposition 2.1Ža.. This shows that the map Ž4.9. is surjective. Step 5. By an argument similar to Step 4, we also have that the canonical map S mR Hom R Ž X , K R . ª Hom S Ž S mR X , S mR K R .
Ž 4.11.
is surjective. Step 6. Lastly, we prove that the sequence Ž4.4. is minimal. Assume the contrary. Then there exist s g Hom S Ž S mR X, S mR K R . and t g Hom S Ž S mR K R , S mR Y . such that s (Ž S mR f .(t is the identity map of S mR K R . By Step 4 and Step 5, we can write s s Ý i a i Ž S mR si . and t s Ý j bj Ž S mR t j . for some si g Hom R Ž X, K R ., t j g Hom R Ž K R , Y ., and a i , b j g S . A s w e h a ve s ( Ž S m R f . ( t s id, w e h a ve S mR ŽÝ i Ý j a i bj Ž si ( f (t j .. s 1 in Hom S Ž S mR K R , S mR K R . ( S. This shows that, at least for one Ž i, j ., we have that si ( f (t j g Hom R Ž K R , K R . s R is not contained in the maximal ideal m, because w is local. This shows that Y and X have the common direct summand K R through f, and this is a contradiction.
5. MAIN THEOREM Our main result in this paper is the following. THEOREM 5.1. Let w : Ž R, m . ª Ž S, n . be a flat local homomorphism of Gorenstein local rings, and F s Srm S be the closed fiber. Then we ha¨ e index Ž S . F index Ž R . ? l l Ž F . . Proof. We proceed in two steps. Step 1. We first prove Theorem 5.1 for d s dim F s 0. Let f s
l l Ž F . and r s indexŽ R .. By definition of Loewy length, we have n f ; m S,
and it follows n f r ; m r S. Since there is an epimorphism Srn f r ª Srm r S ( S mR Rrm r , we know that dS Ž Srn f r . G dS Ž Srm r S . by Proposition 2.6. But Corollary 4.6 shows dS Ž Srm r S . s d R Ž Rrm r . ) 0, and this shows that indexŽ S . F indexŽ R . ? l l Ž F ..
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Step 2. Next we prove Theorem 5.1 in general. Let f s l l Ž F .. Then we can find a system of parameters x 1 , x 2 , . . . , x d g F such that l l Ž F . s l l Ž F ., where F s FrŽ x 1 , x 2 , . . . , x d . F. We set S s SrŽ x 1 , x 2 , . . . , x d . S, where x i is a preimage of x i Ž i s 1, 2, . . . , d ., and w
define the local homomorphism c : R ª S as the composite R ª S ª S, where the second arrow is the natural map. Then x 1 , x 2 , . . . , x d is an S-regular sequence and c is flat by Corollary of Theorem 22.5 in w15x. By Step 1, we have indexŽ S . F indexŽ R . ? l l Ž F .. Thus we know that indexŽ S . F indexŽ S . F indexŽ R . ? l l Ž F . s indexŽ R . ? l l Ž F . by Theorem 1.1. COROLLARY 5.2. Let w : Ž R, m . ª Ž S, n . be a flat local homomorphism of Gorenstein local rings and F s Srm S a regular local ring. Then we ha¨ e index Ž R . s index Ž S . . Proof. This is clear because of Theorems 5.1 and 1.1 and the fact
l l Ž F . s 1. In what follows, k denotes a field. EXAMPLE 5.3. Let r and f be positive integers, and set R s k w X x rŽ X r . ; S s k w X , Y x rŽ X r , Y f . . Then, S is R-flat, and the closed fiber is F s k ww Y xxrŽ Y f .. Then indexŽ R . s r, indexŽ S . s r q f y 1, and l l Ž F . s f. Therefore, we have indexŽ S . F indexŽ R . ? l l Ž F ., and equality holds if and only if r s 1 or f s 1. EXAMPLE 5.4. With the same r, f, and R as in Ž5.3., we set S s k ww X, Y xxrŽ X r , X y Y f . ( k ww Y xxrŽ Y f r .. Then, S is flat over R, and we get F s k ww Y xxrŽ Y f . and indexŽ S . s fr. Therefore, we have indexŽ S . s indexŽ R . ? l l Ž F .. We say that a Z-graded k-algebra R s [i R i is homogeneous when R is generated by finite degree one elements over R 0 s k. The Hilbert series HR Ž t . of R is defined by HR Ž t . [ Ý i G 0 dim k R i t i g Zww t xx. It is known that Q R Ž t . [ Ž1 y t . d HR Ž t . g Zw t x, where d s dim R. The degree of Q R Ž t . is denoted by sŽ R .. In w14x, a graded analogue of d-invariants, indices, and generalized Loewy lengths are discussed. Herzog w14x proved that for a homogeneous Gorenstein k-algebra R, we have indexŽ R . s sŽ R . q 1 s l l Ž R ., provided k is infinite. This result is generalized by Ding in w13x.
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EXAMPLE 5.5. Let R and F be Gorenstein homogeneous k-algebras. Then, S [ R mk F is flat over R with the fiber F, and we have indexŽ S . s indexŽ R . q indexŽ F . y 1. Proof. We may extend the base field k if necessary, and may assume that k is infinite by Corollary 5.2. Obviously, we have HS Ž t . s HR Ž t . HF Ž t .. The equality follows from this and Herzog’s theorem.
6. LOEWY LENGTHS OF ARTINIAN MODULES In this section, R is a local ring which is not necessarily Cohen]Macaulay. Let I ; m be an ideal of R. We set G s Gr I R, and let M denote the graded maximal ideal of G. As for G-modules, we only consider graded ones. The Loewy length l l G Ž H . of an artinian G-module is the smallest integer n such that M n H s 0 Žif one prefers local rings, then he or she might consider G M .. For an R-module M and m g M, we denote the initial form of m in Gr I M by in I Ž m.. LEMMA 6.1. F l l R Ž M ..
Let M be an artinian R-module. Then we ha¨ e l l G ŽGr I M .
Proof. There exists some x 1 , . . . , x r g m such that in I Ž x 1 ., . . . , in I Ž x r . generates M. We set l l ŽGr I M . s s q 1. Then, there exists some 1 F i1 , . . . , i s F r and m g M such that in I Ž x i1 . ??? in I Ž x i s . ? in I Ž m. / 0. This shows in Ž x i1 ??? x i sm . s in I Ž x i1 . ??? in I Ž x i s . ? in I Ž m . / 0, and hence x i1 ??? x i sm / 0. LEMMA 6.2.
Let M be an artinian R-module. If I s m, then we ha¨ e
l l G ŽGrm M . s l l R Ž M ..
Proof. Assume that M s Gr m M s 0. Then we have m s Mrm sq 1 M s 0, and hence m s M s 0. From this and Lemma 6.1, the result follows. PROPOSITION 6.3. Let Ž R, m . and Ž S, n . be local rings, and R ª S a flat local homomorphism with the artinian closed fiber F s Srm S. For an artinian R-module M, we ha¨ e
l l R Ž M . q l l Ž F . y 1 F l l S Ž S mR M . F l l R Ž M . ? l l Ž F . . Proof. We set k s Rrm, R9 s Gr m R, M9 s Gr m M, and S9 s Gr m S S. The graded maximal ideals of R9 and S9 are denoted by m9 and n9, respectively. Then, S9 ( F mk R9 is flat over R9, and we have an S9-isomorphism Gr m S Ž S mR M . ( F mk M9. We set f s l l Ž F . and r s l l R Ž M ..
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Then, we have
Ž n9 .
fqry2
Ž F mk M9 . > n fy1 F mk Ž m9 .
ry1
M9 / 0
by Lemma 6.2. Hence, we have
l l S Ž S mR M . G l l S9 Ž Grm S Ž S mR M . . ) f q r y 2 by Lemma 6.1, and the first inequality follows. As we have n r f Ž S mR M . ; m r Ž S mR M . s 0, the second inequality is obvious. COROLLARY 6.4. Let R ª S be a flat homomorphism of artinian local rings with the fiber F. Then, we ha¨ e
l l Ž R. q l l Ž F . y 1 F l l Ž S . F l l Ž R. ? l l Ž F . . Examples can be found in Section 5.
ACKNOWLEDGMENTS The authors are grateful to Professor S. Goto for suggesting to us the possibility of an inequality for indices which involves the fiber ring. Our special thanks are also due to Professor Y. Yoshino and Professor K. Yoshida for valuable advice.
REFERENCES 1. M. Auslander and R. O. Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Mem. ´ Soc. Math. France 38 Ž1989., 5]37. 2. M. Auslander, S. Ding, and Ø. Solberg, Liftings and weak liftings of modules, J. Algebra 156 Ž1993., 273]317. 3. L. L. Avramov, Locally complete intersection homomorphisms and vanishing of Andre´ Quillen homology, in ‘‘Commutative Algebra: Extended Abstracts of an International Conference, Jul. 27]Aug. 1, 1994. Vechta, Germany’’ ŽW. Bruns, J. Herzog, M. Hochster, and U. Vetter, Eds.., pp. 20]24, Vechtaer Universitatsschriften, Vol. 13, 1994. ¨ 4. L. L. Avramov and H.-B. Foxby, Locally Gorenstein homomorphisms, Amer. J. Math. 114 Ž1992., 1007]1047. 5. L. L. Avramov and H.-B. Foxby, Cohen-Macaulay properties of ring homomorphisms, Ad¨ . Math., to appear. 6. L. L. Avramov, H.-B. Foxby, and B. Herzog, Structure of local homomorphisms, J. Algebra 164 Ž1994., 121]145. 7. W. Bruns and J. Herzog, ‘‘Cohen-Macaulay Rings,’’ Cambridge Univ. Press, Cambridge, UK, 1993.
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8. D. A. Buchsbaum and D. Eisenbud, What makes a complex exact?, J. Algebra 25 Ž1973., 259]268. 9. D. A. Buchsbaum and D. S. Rim, A generalized Koszul complex. II. Depth and multiplicity, Trans. Amer. Math. Soc. 3 Ž1964., 197]225. 10. S. Ding, Cohen-Macaulay approximation and multiplicity, J. Algebra 153 Ž1992., 271]288. 11. S. Ding, A note on the index of Cohen-Macaulay rings, Comm. Algebra 1 Ž1993., 53]71. 12. S. Ding, The associated graded ring and the index of a Gorenstein local ring, Proc. Amer. Math. Soc. 120 Ž1994., 1029]1033. 13. S. Ding, Auslander’s d-invariants of Gorenstein local rings, Proc. Amer. Math. Soc. 122 Ž1994., 649]656. 14. J. Herzog, On the index of a homogeneous Gorenstein ring, Contemp. Math. 159 Ž1994., 95]102. 15. H. Matsumura, ‘‘Commutative Ring Theory,’’ Cambridge Univ. Press, Cambridge, UK, 1986. 16. R. Y. Sharp, Finitely generated modules of finite injective dimension over certain Cohen-Macaulay rings, Proc. London Math. Soc. (3) 25 Ž1971., 303]328. 17. A. Shida, On indices of local rings along local homomorphism, Comm. Algebra, in press. 18. Y. Yoshino, ‘‘Cohen-Macaulay Modules over Cohen-Macaulay Rings,’’ Cambridge Univ. Press, Cambridge, UK, 1990. 19. Y. Yoshino, Cohen-Macaulay approximations, in ‘‘Proc. 4th Symposium on Representation Theory of Algebras, Izu, Japan, 1993,’’ pp., 119]138. wIn Japanesex 20. Y. Yoshino, On the higher delta-invariants of a Gorenstein local ring, Proc. Amer. Math. Soc., in press.
JOURNAL OF ALGEBRA ARTICLE NO.
Some Remarks on Index and Generalized Loewy Length of a Gorenstein Local Ring Mitsuyasu Hashimoto* and Akira Shida† Department of Mathematics, School of Science, Nagoya Uni¨ ersity, Chikusa-ku, Nagoya 464-01, Japan Communicated by Mel¨ in Hochster Received March 29, 1995
1. INTRODUCTION Throughout this paper, the word ‘‘ring’’ will mean commutative noetherian ring with 1. Let R be a Gorenstein local ring. M. Auslander introduced the notion of a d-invariant d R Ž M . for a finitely generated R-module M. It is defined as the smallest integer m such that there exists an epimorphism X [ R m ª M with X a maximal Cohen]Macaulay module which has no non-zero free summand. In studying the d-invariant, Ding w10]13x studied the notion of index. The index of R, denoted by indexŽ R ., is defined as the smallest positive integer such that d R Ž Rrm n . ) 0. He compared indexŽ R . with the generalized Loewy length l l Ž R . of R, which is defined as the minimum of the Loewy lengths of Rrx R for all systems of parameters x of R, and he conjectured Žand proved for some special cases. that indexŽ R . s l l Ž R . in general Žsee Definition 2.7 and Remark 3.1.. In this paper, we first remark that the generalized Loewy length is not stable even under a finite ´ etale extension, but stable under completion ŽSection 3.. We need at least to assume that the residue field of R is infinite to study Ding’s conjecture. Next, we prove that the minimal Cohen]Macaulay approximation, which is indispensable when we study d-invariants, of a finitely generated module M over a Cohen]Macaulay local ring R is preserved by an extension by * Current address: Nagoya University College of Medical Technology, 1-1-20 Daikominami, Higashi-ku, Nagoya 461, Japan. E-mail address: [email protected]. † E-mail address: [email protected]. 150 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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a Gorenstein local homomorphism R ª S under the assumption ToriR Ž S, M . s 0 Ž i ) 0. ŽProposition 4.3.. If R is Gorenstein moreover, then the higher delta invariants are also preserved ŽCorollary 4.6.. This unifies two known important cases: the flat case and the case RS s RrxR with x both R-regular and M-regular. We note that many of the notions and the results on flat morphisms have been generalized to the situation of finite flat dimension w4, 5, 3x. Let Ž R, m . and Ž S, n . be Gorenstein local rings, and w : R ª S be a local homomorphism of finite flat dimension Ži.e., the flat dimension of S as an R-module is finite.. Recently, the second author proved the following. THEOREM 1.1 w17, Ž3.7.x.
Let w : R ª S be as abo¨ e. Then we ha¨ e
index Ž R . F index Ž S . . This can be seen as a sort of flat descent Žgeneralized to the finite flat dimension context.. A good property ‘‘index F n’’ of S is inherited by R for any n. In Section 5, assuming S is R-flat, we prove indexŽ S . F indexŽ R . ? l l Ž F . Ž F [ Srm S ., which can be seen as a counterpart of the theorem above. This is our main theorem. Furthermore, this inequality shows that if the closed fiber F is a regular local ring, we have indexŽ R . s indexŽ S . as might be expected. We cannot expect that the fiber ring F s Srm S has a small index even if S has a small index. This can be seen by the following example: R s k ww x t xx ; k ww x xx s S, where t is an arbitrary positive integer. However, if S is artinian, then we have an inequality: l l Ž R . q l l Ž F . y 1 F l l Ž S . F l l Ž R . ? l l Ž F .. This will be proved in Section 6 for artinian rings which are not necessarily Gorenstein.
2. PRELIMINARIES Unless otherwise specified, we assume that Ž R, m , k . is a Cohen]Macaulay local ring with the canonical module K R . By Rˆ we mean the completion of R with respect to the maximal ideal m. For an ˆ denotes Rˆ mR M. For a proper ideal I of R, we denote R-module M, M the associated graded module [i G 0 I i MrI iq1 M by Gr I M. Let Ž S, n . be a local ring, and w : R ª S a homomorphism. We say that w is local when w Ž m . ; n. PROPOSITION 2.1. Let R be a Cohen]Macaulay local ring, X a maximal Cohen]Macaulay R-module, and Y Ž resp. Z . a finitely generated R-module of finite injecti¨ e Ž resp. projecti¨ e . dimension. Then we ha¨ e Ža. Ext iR Ž X, Y . s 0 for i ) 0; Žb. ToriR Ž Z, X . s 0 for i ) 0.
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Proof. Part Ža. is noted in w18x. Part Žb. is a slight generalization of w16, Ž2.6.x, and follows easily from w8x. Let M be a finitely generated R-module. A sequence of finitely generated R-modules f
g
0 ª Yª XªM ª 0
Ž 2.2.
is called a Cohen]Macaulay approximation if X is a maximal Cohen]Macaulay R-module, Y is of finite injective dimension, and the sequence is exact. The Cohen]Macaulay approximation Ž2.2. is said to be minimal when X and Y have no non-zero direct summand in common through f. It is said to be right minimal when g is right minimal, that is, for any non-isomorphic map w : X ª X, we have g ( w / g. The next lemma and its corollary seem to be well known, but we state them with proofs. The complete case Žwhich is essential . is found in w19x. LEMMA 2.3.
Let f
g
A [ 0 ª Yª XªM ª 0 be a sequence of finitely generated R-modules. Then, the following conditions are equi¨ alent. Žrmin. A is a right minimal Cohen]Macaulay approximation of M. Žmin. A is a minimal Cohen]Macaulay approximation of M. $
ˆ is a right minimal Cohen]Macaulay approximation of M. ˆ Žrmin. A $
ˆ is a minimal Cohen]Macaulay approximation of M. ˆ Žmin. A $
Proof. It is easy to see that A is a Cohen]Macaulay approximation if and only if A is a Cohen]Macaulay approximation. Thus, the problem is the minimality. Žrmin. « Žmin.. Assume that X s X 0 [ X 1 and 0 / X 0 ; Im f. We define w : X ª X by w Ž x 0 q x 1 . s x 1 for x 0 g X 0 and x 1 g X 1. Then, we have g ( w s$ g, and w is not isomorphic. Žmin. « Žmin.. See Proposition 4.3 below. $ $ ˆ s A. Assume that Žmin. « Žrmin.. We may assume Rˆ s R and A w g End R Ž X . is not an isomorphism, and g ( w s g. We have X / 0, and hence M / 0 by minimality of A. We set L to be the sub-R-algebra of End R Ž X . generated by R and w . Note that L is commutative, and is module finite over R. As w preserves the kernel of g, we may and shall regard Y and M as L-modules so that f and g are L-linear. Note that w g L acts as an identity map on M. By Nakayama’s lemma, w is not contained in the radical of L, since w Ž M . s M and M / 0. As w is not a
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unit in L Žsince w is not an isomorphism on X ., we have that L is not local. It follows that L has a non-trivial idempotent, say e / 0, 1, since R is henselian. With replacing e with 1 y e if necessary, we may assume that the image of e in LrŽ1 y w . L, which is local, is a unit. As e acts as an automorphism on M, it is easy to see that X 0 [ ImŽ1 y e . s Ker e is a non-zero direct summand of X which is contained in Im f. $ Žrmin. « Žrmin.. Assume that w g End R Ž X . is non-isomorphic and that g ( w s g. Then, w ˆ is non-isomorphic and ˆg ( wˆ s ˆg. COROLLARY 2.4. Let M be a finitely generated R-module. Then, there exists a minimal Cohen]Macaulay approximation of M, uniquely up to isomorphism. Proof. A Cohen]Macaulay approximation of M exists, see w1x. Removing non-zero common direct summands through f if any, we obtain a minimal one from this. The uniqueness of the right minimal Cohen]Macaulay approximation is shown by a standard comparison argument in w1x. For an R-module M, we define the f-rank of M, denoted by f-rank R M, to be the number max i < R i is a direct summand of M 4 . DEFINITION 2.5. Let R be a Cohen]Macaulay local ring with K R , and let 0 ª Y ª X ª M ª 0 be the minimal Cohen]Macaulay approximation of M over R. We define the d-in¨ ariant of M, denoted by d R Ž M ., as f-rank R X. And we define the index of a Gorenstein local ring R, denoted by indexŽ R ., as the integer min n < d R Ž Rrm n . / 04 . For a nonnegative integer n, the d-invariant of the nth syzygy V nR Ž M . of M is denoted by d Rn Ž M ., and we call it the nth d-in¨ ariant of M. PROPOSITION 2.6 w10x. modŽ R .. Then
Let R be a Gorenstein local ring and M Ž/ 0. g
Ža. If pd R M is finite, then d R Ž M . s m Ž M .Ž) 0., where m Ž M . is the minimal number of generators of M. Žb. If M ª N ª 0 is exact, then d R Ž M . G d R Ž N .. Next we define the generalized Loewy length. DEFINITION 2.7. Let M be an R-module of finite length. The Loewy length of M, denoted by l l R Ž M ., is the smallest integer n such that m n M s 0. We define the generalized Loewy length of R, which is also denoted by l l Ž R ., as the minimum of all integers l l R Ž RrŽx.., where x is a system of parameters of R.
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3. REMARKS ON GENERALIZED LOEWY LENGTH Remark 3.1. Ding proved that indexŽ R . F l l Ž R . for any Gorenstein local ring R w11x. He conjectured that indexŽ R . s l l Ž R . for the arbitrary Gorenstein local ring R. He claimed that the conjecture is true when the associated graded ring Gr m R of R is a Cohen]Macaulay ring w12x. He also claimed that the conjecture is true when R is gradable and depth Gr m R G dim R y 1 w13x. These are certainly true when Rrm is infinite. If we drop the condition on the residue field, there is a counterexample to the conjecture. EXAMPLE 3.2. Let k s F 2 , and K s F4 , where Fq is the q-element field. When we set S s k ww x, y xx, f s xy Ž x q y . g S, and R s SrŽ f ., then we have 4 s l l Ž R . ) l l Ž K mk R . s index Ž K mk R . s index Ž R . s 3. Proof. As the rings in consideration are hypersurfaces, we have that indices equal to multiplicities w10x, and we have indexŽ K mk R . s indexŽ R . s 3. Let v g K _ k, and set z s x y v y. Then, we have l l ŽŽ K mk R .rŽ z .. s 3. Hence, we have l l Ž K mk R . s 3. As we have l l Ž RrŽ x y y 2 .. s 4, we have l l Ž R . F 4. It remains to show that l l Ž R . ) 3. Assume the contrary. Then, there exists some R-regular element z g m such that m 3 ; Ž z ., where m s Ž x, y . R. We set r s max i < z g m i 4 . It is easy to see that r F 2. If r s 2, then zR is annihilated by m 2 , where R s Rrm 4 . Hence, we have 3 s l R Ž Rrm 2 . G l R Ž Rz . G l R Ž m 3 R . s l R Ž m 3rm 4 . s 3. This shows zR s m 3 R, and hence z g m 3. This contradicts r s 2. Consider the case r s 1. Take any preimage z g S of z. Letting an appropriate element in GL 2 Ž k . act on S, we may assume that z s x y g Ž x, y . with g g ŽŽ x, y . S . 2 without loss of generality. As RrŽ z . is a hypersurface, it suffices to show that l R Ž RrŽ z .. G 4 to lead to a contradiction. When we set R9 s SrŽ z ., then R9 is a discrete valuation ring, and RrŽ z . s R9rŽ f ., where f s xy Ž x q y . is the image of f in R9. On the other hand, we have y g Ž x, y . R9, x q y g Ž x, y . R9, and x s g g ŽŽ x, y . R9. 2 . Hence, we have f g ŽŽ x, y . R9. 4 and we have l Ž R9rŽ f .. G 4. As we have seen, the generalized Loewy length is not stable under a finite ´ etale extension. However, it is stable under completion.
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LEMMA 3.3. We ha¨ e l l Ž R . s l l Ž Rˆ.. Proof. If x is a system of parameters of R such that l l Ž R . s ˆ Rˆ. s l l R Ž Rrx R . s l l Ž R .. l l R Ž Rrx R ., then we have l l Ž Rˆ. F l l RˆŽ Rrx So it suffices to show l l Ž R . F l l Ž Rˆ.. Let n s l l Ž Rˆ., and take a system of parameters Ž ˆ x1, ˆ x2 , . . . , ˆ x d . ; Rˆ such that m n Rˆ ; Ž ˆ x1, ˆ x2 , . . . , ˆ x d ., where d s dim R. Then for each ˆ xi Ž i s 1, 2, . . . , d ., we can choose an element x i g R such that x i y ˆ xi g ˆ We claim that m n ; Ž x 1 , x 2 , . . . , x d .. In fact since m n Rˆ ; m nq 1 R. ˆ we have Žˆ x1, ˆ x2 , . . . , ˆ x d . ; Ž x 1 , x 2 , . . . , x d . Rˆ q m nq 1 R, m n Rˆ ;
ˆ F ž Ž x 1 , x 2 , . . . , x d . Rˆ q m nq i Rˆ/ s Ž x 1 , x 2 , . . . , x d . R. i)0
Hence, we have m n s m n Rˆ l R ; Ž x 1 , x 2 , . . . , x d . Rˆ l R s Ž x 1 , x 2 , . . . , x d . as desired.
4. A GORENSTEIN HOMOMORPHISM AND MINIMAL COHEN]MACAULAY APPROXIMATIONS Let w : R ª S be a flat local homomorphism of Gorenstein local rings. If 0 ª Y ª X ª M ª 0 is a Žminimal. Cohen]Macaulay approximation over R, then so is 0 ª S mR Y ª S mR X ª S mR M ª 0. In particular, the Žhigher. d-invariants are preserved by this base extension. In this section, we generalize this to the context of morphisms of finite flat dimension ŽProposition 4.3, Corollary 4.6.. The case S s RrxR with x both R- and M-regular w2, Ž5.1.; 20, Ž1.8.x, as well as the flat case w20, Ž1.5., Ž1.7.x, has been well known. First, we introduce the notion of a Gorenstein local homomorphism Žsee w4x.. Let w : Ž R, m . ª Ž S, n . be a local homomorphism. We say that w is a Gorenstein local homomorphism if w is of finite flat dimension Ži.e., fd R S - `., and m Ri s mSiqs for some s and arbitrary i G 0, where m Ri s dim k Ext Ri Ž k, R . is the ith Bass number. Next, we define a Cohen factorization which was introduced in w6x. DEFINITION 4.1. Let w : Ž R, m . ª Ž S, n . be a local homomorphism of local rings. We say w is factorizable if it can be decomposed as w s st with t : R ª T flat and TrmT a regular local ring, s : T ª S surjective, and T a complete local ring; often we shall refer to such a situation by saying that w s st is a Cohen factorization.
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It is known that if S is a complete local ring, then there exists a Cohen factorization Žsee w6, Ž1.1.x.. Let w : R ª S be a local homomorphism, and w s st its Cohen factorization. Then, w is of finite flat dimension if and only if so is s w6, Ž3.3.x. Note also that, if w is module finite, then we have fd R S - ` m ToriR Ž S, Rrm . s 0 for i 4 0 m pd R S - `. By w4, Ž4.2., Ž4.6.x, w is Gorenstein if and only if so is s . LEMMA 4.2. Let w : R ª S be a local homomorphism of Cohen]Macaulay local rings. We assume that fd R S - `. Let X be a maximal Cohen]Macaulay R-module. Then we ha¨ e ToriR Ž S, X . s 0 for i ) 0. We also ha¨ e S mR X is a maximal Cohen]Macaulay S-module. Proof. We may assume that S is complete. If S is R-flat, then the assertion is obvious. Consider the general case. Take a Cohen factorization w s st . As the lemma is true for t , we may replace w by s , and we may assume that w is surjective. Now, the vanishing ToriR Ž S, X . s Ž i ) 0. is Proposition 2.1Žb.. As for the last assertion, the same proof in w17, Ž3.1.x works. PROPOSITION 4.3. Let w : R ª S be a Gorenstein local homomorphism of Cohen]Macaulay local rings, and M a finitely generated R-module. Furthermore, we assume that R has the canonical module K R , and ToriR Ž S, M . s 0 f
for all i ) 0. If 0 ª Y ª X ª M ª 0 is the minimal Cohen]Macaulay approximation of M o¨ er R, then the sequence SmR f
S mR X ª S mR M ª 0
6
0 ª S mR Y
Ž 4.4.
is the minimal Cohen]Macaulay approximation of S mR M o¨ er S. Remark 4.5. When w is surjective and S s RrI, then, for r G 0, we have ToriR Ž S, M . s 0 Ž i ) r . if and only if depth R Ž I, M . G pd R S y r Ždepth sensitivity of perfect ideals, see w9x.. In particular, the Tor-independence assumption ToriR Ž S, M . s 0 Ž i ) 0. above is equivalent to depth R Ž I, M . s pd R S. Generalizing this to the imperfect case, the second author proved the following, which we only state the result here: Let R be a Žnot necessarily Cohen]Macaulay. ring, I an ideal of R of finite projective dimension, and S s RrI. If M is a finitely generated R-module with MrIM / 0, then we have sup i < ToriR Ž S, M . / 0 4 ssup pd R pSp ydepth R pMp < pg supp Ž MrIM . 4 . COROLLARY 4.6. In the proposition, assume moreo¨ er that R is Gorenstein. Then we ha¨ e f-rank S Ž S mR X . s f-rank R X . In particular, we ha¨ e
dSn Ž S
mR M . s
d Rn Ž M ..
Ž 4.7.
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Proof. The equality Ž4.7. follows from the proposition and w17, Ž3.1.x. As we have ToriR Ž S, M . s 0 Ž i ) 0. by assumption, we have V Sn Ž S mR M . ( S mR V nR Ž M ., and we have dSn Ž S mR M . s d Rn Ž M .. Proof of Proposition 4.3. Since it is sufficient to prove that ˆ Rf Sm
Sˆ mR X ª Sˆ mR M ª 0
6
0 ª Sˆ mR Y
is the minimal Cohen]Macaulay approximation of Sˆ mR M, we may assume that S is complete. We proceed in several steps. Step 0. Note that ToriR Ž S, K R . s 0 for i ) 0 by Lemma 4.2. We have that S mR K R ( K S , since w is Gorenstein w4, Ž5.1.x. Step 1. Since ToriR Ž S, M . s 0 for i ) 0 by assumption and RŽ Tori S, X . s 0 by Lemma 4.2, we have ToriR Ž S, Y . s 0 for i ) 0. Step 2. We show that S mR Y is of finite injective dimension over S. First note that a module Y is of finite injective dimension if and only if there exists an exact sequence wn
w ny1
w1
w0
0 ª In ª Iny1 ª ??? ª I0 ª Y ª 0
Ž 4.8.
such that I j is a finite direct sum of K R Ž j s 0, 1, . . . , n.. As the modules in Ž4.8. are Tor-independent of S by Step 0 and Step 1, the sequence ???
S mR Y
0
6
XmR w 0
SmR w ny1
6
S mR Iny1 6
S mR I0
6
SmR w 1
SmR w n
6
0 ª S mR In
is exact. Here S mR I j is a finite direct sum of K S by Step 0. Thus, we conclude that S mR Y is of finite injective dimension as an S-module. Step 3. We know from Step 2 that S mR Y is of finite injective dimension. Since S mR X is a maximal Cohen]Macaulay S-module by Lemma 4.2, and Ž4.4. is an exact sequence by Tor1R Ž S, M . s 0, we have that Ž4.4. is a Cohen]Macaulay approximation of S mR M over S. Step 4. Next, we show that the canonical homomorphism S mR Hom R Ž K R , Y . ª Hom S Ž S mR K R , S mR Y .
Ž 4.9.
given by a m f ¬ a ? Ž S mR f . for a g S and f g Hom R Ž K R , Y . is surjective. As this is obvious when S is R-flat, we may assume that w is surjective Žconsider the Cohen-factorization of w . so that S s RrI for a proper ideal I of R of finite projective dimension. Note that 0 ª I mR Y ª Y ª S mR Y ª 0 is exact since i ) 0.
Tor1R Ž S, Y
. s 0, and that
ToriR Ž I, Y
.(
Ž 4.10. R Ž Toriq1 S, Y
. s 0 for
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Let F be a finite free resolution of I. Then, as we have that F mR Y ª I mR Y is quasi-isomorphic and that each term of F mR Y is of finite injective dimension, we have that the injective dimension of I mR Y is finite. From the exact sequence Ž4.10., we obtain an exact sequence Hom R Ž K R , Y . ª Hom R Ž K R , S m Y . ª 0 s Ext 1R Ž K R , I mR Y . by Proposition 2.1Ža.. This shows that the map Ž4.9. is surjective. Step 5. By an argument similar to Step 4, we also have that the canonical map S mR Hom R Ž X , K R . ª Hom S Ž S mR X , S mR K R .
Ž 4.11.
is surjective. Step 6. Lastly, we prove that the sequence Ž4.4. is minimal. Assume the contrary. Then there exist s g Hom S Ž S mR X, S mR K R . and t g Hom S Ž S mR K R , S mR Y . such that s (Ž S mR f .(t is the identity map of S mR K R . By Step 4 and Step 5, we can write s s Ý i a i Ž S mR si . and t s Ý j bj Ž S mR t j . for some si g Hom R Ž X, K R ., t j g Hom R Ž K R , Y ., and a i , b j g S . A s w e h a ve s ( Ž S m R f . ( t s id, w e h a ve S mR ŽÝ i Ý j a i bj Ž si ( f (t j .. s 1 in Hom S Ž S mR K R , S mR K R . ( S. This shows that, at least for one Ž i, j ., we have that si ( f (t j g Hom R Ž K R , K R . s R is not contained in the maximal ideal m, because w is local. This shows that Y and X have the common direct summand K R through f, and this is a contradiction.
5. MAIN THEOREM Our main result in this paper is the following. THEOREM 5.1. Let w : Ž R, m . ª Ž S, n . be a flat local homomorphism of Gorenstein local rings, and F s Srm S be the closed fiber. Then we ha¨ e index Ž S . F index Ž R . ? l l Ž F . . Proof. We proceed in two steps. Step 1. We first prove Theorem 5.1 for d s dim F s 0. Let f s
l l Ž F . and r s indexŽ R .. By definition of Loewy length, we have n f ; m S,
and it follows n f r ; m r S. Since there is an epimorphism Srn f r ª Srm r S ( S mR Rrm r , we know that dS Ž Srn f r . G dS Ž Srm r S . by Proposition 2.6. But Corollary 4.6 shows dS Ž Srm r S . s d R Ž Rrm r . ) 0, and this shows that indexŽ S . F indexŽ R . ? l l Ž F ..
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Step 2. Next we prove Theorem 5.1 in general. Let f s l l Ž F .. Then we can find a system of parameters x 1 , x 2 , . . . , x d g F such that l l Ž F . s l l Ž F ., where F s FrŽ x 1 , x 2 , . . . , x d . F. We set S s SrŽ x 1 , x 2 , . . . , x d . S, where x i is a preimage of x i Ž i s 1, 2, . . . , d ., and w
define the local homomorphism c : R ª S as the composite R ª S ª S, where the second arrow is the natural map. Then x 1 , x 2 , . . . , x d is an S-regular sequence and c is flat by Corollary of Theorem 22.5 in w15x. By Step 1, we have indexŽ S . F indexŽ R . ? l l Ž F .. Thus we know that indexŽ S . F indexŽ S . F indexŽ R . ? l l Ž F . s indexŽ R . ? l l Ž F . by Theorem 1.1. COROLLARY 5.2. Let w : Ž R, m . ª Ž S, n . be a flat local homomorphism of Gorenstein local rings and F s Srm S a regular local ring. Then we ha¨ e index Ž R . s index Ž S . . Proof. This is clear because of Theorems 5.1 and 1.1 and the fact
l l Ž F . s 1. In what follows, k denotes a field. EXAMPLE 5.3. Let r and f be positive integers, and set R s k w X x rŽ X r . ; S s k w X , Y x rŽ X r , Y f . . Then, S is R-flat, and the closed fiber is F s k ww Y xxrŽ Y f .. Then indexŽ R . s r, indexŽ S . s r q f y 1, and l l Ž F . s f. Therefore, we have indexŽ S . F indexŽ R . ? l l Ž F ., and equality holds if and only if r s 1 or f s 1. EXAMPLE 5.4. With the same r, f, and R as in Ž5.3., we set S s k ww X, Y xxrŽ X r , X y Y f . ( k ww Y xxrŽ Y f r .. Then, S is flat over R, and we get F s k ww Y xxrŽ Y f . and indexŽ S . s fr. Therefore, we have indexŽ S . s indexŽ R . ? l l Ž F .. We say that a Z-graded k-algebra R s [i R i is homogeneous when R is generated by finite degree one elements over R 0 s k. The Hilbert series HR Ž t . of R is defined by HR Ž t . [ Ý i G 0 dim k R i t i g Zww t xx. It is known that Q R Ž t . [ Ž1 y t . d HR Ž t . g Zw t x, where d s dim R. The degree of Q R Ž t . is denoted by sŽ R .. In w14x, a graded analogue of d-invariants, indices, and generalized Loewy lengths are discussed. Herzog w14x proved that for a homogeneous Gorenstein k-algebra R, we have indexŽ R . s sŽ R . q 1 s l l Ž R ., provided k is infinite. This result is generalized by Ding in w13x.
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EXAMPLE 5.5. Let R and F be Gorenstein homogeneous k-algebras. Then, S [ R mk F is flat over R with the fiber F, and we have indexŽ S . s indexŽ R . q indexŽ F . y 1. Proof. We may extend the base field k if necessary, and may assume that k is infinite by Corollary 5.2. Obviously, we have HS Ž t . s HR Ž t . HF Ž t .. The equality follows from this and Herzog’s theorem.
6. LOEWY LENGTHS OF ARTINIAN MODULES In this section, R is a local ring which is not necessarily Cohen]Macaulay. Let I ; m be an ideal of R. We set G s Gr I R, and let M denote the graded maximal ideal of G. As for G-modules, we only consider graded ones. The Loewy length l l G Ž H . of an artinian G-module is the smallest integer n such that M n H s 0 Žif one prefers local rings, then he or she might consider G M .. For an R-module M and m g M, we denote the initial form of m in Gr I M by in I Ž m.. LEMMA 6.1. F l l R Ž M ..
Let M be an artinian R-module. Then we ha¨ e l l G ŽGr I M .
Proof. There exists some x 1 , . . . , x r g m such that in I Ž x 1 ., . . . , in I Ž x r . generates M. We set l l ŽGr I M . s s q 1. Then, there exists some 1 F i1 , . . . , i s F r and m g M such that in I Ž x i1 . ??? in I Ž x i s . ? in I Ž m. / 0. This shows in Ž x i1 ??? x i sm . s in I Ž x i1 . ??? in I Ž x i s . ? in I Ž m . / 0, and hence x i1 ??? x i sm / 0. LEMMA 6.2.
Let M be an artinian R-module. If I s m, then we ha¨ e
l l G ŽGrm M . s l l R Ž M ..
Proof. Assume that M s Gr m M s 0. Then we have m s Mrm sq 1 M s 0, and hence m s M s 0. From this and Lemma 6.1, the result follows. PROPOSITION 6.3. Let Ž R, m . and Ž S, n . be local rings, and R ª S a flat local homomorphism with the artinian closed fiber F s Srm S. For an artinian R-module M, we ha¨ e
l l R Ž M . q l l Ž F . y 1 F l l S Ž S mR M . F l l R Ž M . ? l l Ž F . . Proof. We set k s Rrm, R9 s Gr m R, M9 s Gr m M, and S9 s Gr m S S. The graded maximal ideals of R9 and S9 are denoted by m9 and n9, respectively. Then, S9 ( F mk R9 is flat over R9, and we have an S9-isomorphism Gr m S Ž S mR M . ( F mk M9. We set f s l l Ž F . and r s l l R Ž M ..
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Then, we have
Ž n9 .
fqry2
Ž F mk M9 . > n fy1 F mk Ž m9 .
ry1
M9 / 0
by Lemma 6.2. Hence, we have
l l S Ž S mR M . G l l S9 Ž Grm S Ž S mR M . . ) f q r y 2 by Lemma 6.1, and the first inequality follows. As we have n r f Ž S mR M . ; m r Ž S mR M . s 0, the second inequality is obvious. COROLLARY 6.4. Let R ª S be a flat homomorphism of artinian local rings with the fiber F. Then, we ha¨ e
l l Ž R. q l l Ž F . y 1 F l l Ž S . F l l Ž R. ? l l Ž F . . Examples can be found in Section 5.
ACKNOWLEDGMENTS The authors are grateful to Professor S. Goto for suggesting to us the possibility of an inequality for indices which involves the fiber ring. Our special thanks are also due to Professor Y. Yoshino and Professor K. Yoshida for valuable advice.
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