Stability of Gorenstein categories

Transcription

1 J. London Math. Soc. (2) 77 (28) C 28 London Mathematical Society doi:1.1112/jlms/jdm124 Stability of Gorenstein categories Sean Sather-Wagstaff, Tirdad Sharif and Diana White Abstract We show that an iteration of the procedure used to define the Gorenstein projective modules over a commutative ring R yields exactly the Gorenstein projective modules. Specifically, given an exact sequence of Gorenstein projective R-modules G = G 2 1 G G 1 G G such that the complexes Hom R(G, H) and Hom R(H, G) are exact for each Gorenstein projective R-module H, the module Coker( 1 G ) is Gorenstein projective. The proof of this result hinges upon our analysis of Gorenstein subcategories of abelian categories. Introduction Let R be a commutative ring. Building from Auslander and Bridger s work [1, 2] on modules of finite G-dimension, Enochs and Jenda [8] and Holm [14] introduced and studied the Gorenstein projective R-modules as the modules of the form Coker( 1 P ) for some exact sequence of projective R-modules P = P 2 P P 1 1 P P such that the complex Hom R (P, Q) is exact for each projective R-module Q. The class of Gorenstein projective R-modules is denoted by G(P(R)). In this paper, we investigate the modules that arise from an iteration of this construction. To wit, let G 2 (P(R)) denote the class of R-modules M for which there exists an exact sequence of Gorenstein projective R-modules G = G 2 G G 1 1 G G such that the complexes Hom R (G, H) and Hom R (H, G) are exact for each Gorenstein projective R-module H and M Coker( 1 G ). One checks readily that there is a containment G(P(R)) G 2 (P(R)). We answer a question from the folklore of this subject by verifying that this containment is always an equality. This is a consequence of Corollary 4.1, as is the dual version for Gorenstein injective R-modules; see Example 5.3. Theorem A. If R is a commutative ring, then G(P(R)) = G 2 (P(R)). The proof of this result is facilitated by the consideration of a more general situation. Starting with a class of R-modules W, we consider the associated full subcategory G(W) of the category of R-modules with objects defined as above; see Definition 4.1. Section 4 is devoted to the category-theoretic properties of G(W), those needed for the proof of Theorem A and others. For instance, the next result is contained in Proposition 4.11 and Theorem Received 21 March 27; published online 21 February Mathematics Subject Classification 13C5, 13D2, 13D7, 18G1, 18G15. Tirdad Sharif is supported by a grant from IPM (No ).

2 482 S. SATHER-WAGSTAFF, T. SHARIF AND D. WHITE Theorem B. Assume that Ext i R(W, W )= for all W, W W and all i 1. The Gorenstein subcategory G(W) is an exact category, and it is closed under kernels of epimorphisms (or cokernels of monomorphisms) if W is so. Most of the paper focuses on subcategories of an abelian category A. The reader is encouraged to keep certain module-categories in mind. Specific examples are provided in Section 3, and we apply our results to these examples in Section Categories and resolutions Here we set some notation and terminology for use throughout this paper. Definition 1.1. In this work A is an abelian category. We use the term subcategory for a full additive subcategory that is closed under isomorphisms. Write P = P(A) and I = I(A) for the subcategories of projective and injective objects in A, respectively. A subcategory X of A is exact if it is closed under direct summands and extensions. We fix subcategories X, Y, W, and V of A such that W X and V Y. Write X Yif Ext 1 A (X, Y ) = for each object X in X and each object Y in Y. For an object A in A, write A Y if Ext 1 A (A, Y ) = for each object Y in Y (and X A if Ext1 A (X, A) = for each object X in X ). We say that W is a cogenerator for X if, for each object X in X, there exists an exact sequence in X X W X such that W is an object in W. The subcategory W is an injective cogenerator for X if W is a cogenerator for X and X W. We say that V is a generator for Y if, for each object Y in Y, there exists an exact sequence in Y Y V Y such that V is an object in V. The subcategory V is a projective generator for Y if V is a generator for Y and V Y. Definition 1.2. An A-complex is a sequence of homomorphisms in A A = A n+1 A n A n A n 1 A n 1 such that A n 1 A n = for each integer n; the nth homology object of A is H n (A) = Ker( n A )/ Im( n+1). A We frequently identify objects in A with complexes concentrated in degree. Fix an integer i. The ith suspension of a complex A, denoted by Σ i A, is the complex with (Σ i A) n = A n i and Σi A n =( 1) i n i. A The hard truncation A i is the complex A i = A i+2 A i+1 A i+1 A i and the hard truncations A >i, A i, and A <i are defined similarly. The complex A is Hom A (X, )-exact if the complex Hom A (X, A) is exact for each object X in X. Dually, it is Hom A (, X )-exact if the complex Hom A (A, X) is exact for each object X in X.

3 STABILITY OF GORENSTEIN CATEGORIES 483 Definition 1.3. Let A, A be A-complexes. The Hom-complex Hom A (A, A ) is the complex of abelian groups defined as Hom A (A, A ) n = p Hom A(A p,a p+n) with nth differential Hom A(A,A ) n given by {f p } { p+nf A p ( 1) n f n 1 p A }. A morphism is an element of Ker( Hom A(A,A ) ) and α is null-homotopic if it is in Im( Hom A(A,A ) 1 ). Given a second morphism α : A A we say that α and α are homotopic if the difference α α is nullhomotopic. The morphism α is a homotopy equivalence if there is a morphism β : A A such that βα is homotopic to id A and αβ is homotopic to id A. The complex A is contractible if the identity morphism id A is null-homotopic. When A is contractible, it is exact, as is each of the complexes Hom A (A, N) and Hom A (M,A) for all objects M and N in A. A morphism of complexes α: A A induces homomorphisms H n (α): H n (A) H n (A ), and α is a quasi-isomorphism when each H n (α) is an isomorphism. The mapping cone of α is the complex Cone(α) defined as Cone(α) n = A n A n 1 with nth differential ( ) n Cone(α) A = n α n 1 n 1 A. This definition gives a degreewise split exact sequence A Cone(α) ΣA. Further, the morphism α is a quasi-isomorphism if and only if Cone(α) is exact. Finally, if id A is the identity morphism for A, then Cone(id A ) is contractible. Definition 1.4. A complex X is bounded if X n = for n. When X n ==H n (X) for all n>, the natural morphism X M =H (X) is a quasi-isomorphism. In this event, X is an X -resolution of M if each X n is an object in X, and the following exact sequence is the augmented X -resolution of M associated to X: X + = X 2 X X 1 1 X M. Instead of writing P-resolution we will write projective resolution. The X -projective dimension of M is the quantity X -pd(m) = inf{sup{n X n } X is an X -resolution of M}. The objects of X -projective dimension are exactly the objects of X.Weset res X = the subcategory of objects M of A with X -pd(m) <. We define Y-coresolutions and Y-injective dimension dually. The augmented Y-coresolution associated to a Y-coresolution Y is denoted + Y, and the Y-injective dimension of M is denoted Y- id(m). We set cores Ŷ = the subcategory of objects N of A with Y- id(n) <. Definition 1.5. An X -resolution X is X -proper (or simply proper) if the augmented resolution X + is Hom A (X, )-exact. We set res X = the subcategory of objects of A admitting a proper X -resolution. One checks readily that res X is additive. If X is an object in X, then the complex X is a proper X -resolution of X ; hence X is in res X and so X res X. Projective resolutions are always P-proper, and so A has enough projectives if and only if res P = A. IfM is an object in A that admits an X -resolution X M and a projective resolution P M, then there exists a quasi-isomorphism P X. Proper coresolutions are defined dually, and we set cores Ỹ = the subcategory of objects of A admitting a proper Y-coresolution.

4 484 S. SATHER-WAGSTAFF, T. SHARIF AND D. WHITE Again, cores Ỹ is additive and Y cores Ỹ. Injective coresolutions are always I-proper, and so A has enough injectives if and only if cores Ĩ = A. If N is an object in A that admits a Y-coresolution N Y and an injective resolution N I, then there exists a quasi-isomorphism Y I. The next lemmata are standard or have standard proofs; for Lemma 1.6 see the proof of [3, Theorem 2.3], for Lemma 1.7 see the proof of [3, Proposition 2.1], for Lemma 1.8 repeatedly apply Definition 1.1, and for the horseshoe lemma, Lemma 1.9, see [9, proof of Lemma 8.2.1]. Lemma 1.6. Let A 1 A 2 A 3 be an exact sequence in A. (a) If A 3 W, then A 1 Wif and only if A 2 W.IfA 1 Wand A 2 W, then A 3 W if and only if the given sequence is Hom A (, W) exact. (b) If V A 1, then V A 2 if and only if V A 3.IfV A 2 and V A 3, then V A 1 if and only if the given sequence is Hom A (V, ) exact. Lemma 1.7. If X Y, then X res Ŷ and cores X Y. Lemma 1.8. If W is an injective cogenerator for X, then every object X in X admits a proper W-coresolution, and so X cores W. IfV is a projective generator for Y, then every object Y in Y admits a proper V-resolution, and so Y res Ṽ. Lemma 1.9. Let A A A be an exact sequence in A. (a) Assume that A and A admit proper X -resolutions X A and X A. If the given sequence is Hom A (X, )-exact, then A is in res X with proper X -resolution X A such that there exists a commutative diagram X ( ) idx X ( id X X ) A A A in which the top row is degreewise split exact and ( ) n X X n f = n. n X (b) Assume that A and A admit proper Y-coresolutions A Y and A Y.Ifthe given sequence is Hom A (, Y)-exact, then A is in cores Ỹ with proper Y-coresolution A Y such that there exists a commutative diagram A A ( idy ( id Y Y ) Y Y in which the bottom row is degreewise split exact and ( ) n Y Y n g = n. ) Y n A

5 STABILITY OF GORENSTEIN CATEGORIES Technical results This section consists of three lemmata which the reader may wish to skip during the first reading. The first result is for use in the proof of Lemma 2.2. Lemma 2.1. Let N N N be an exact sequence in A. Assume that W is an injective cogenerator for X and V is a projective generator for Y. (a) Assume that N is an object in cores X and N is an object in cores W. IfN Wand N W, then N is an object in cores W. (b) Assume that N is an object in res Ỹ, and N is an object in res Ṽ. IfV N and V N, then N is an object in res Ṽ. Proof. We prove part (a); the proof of part (b) is dual. Let X be a proper X -coresolution of N and set N i = Ker( X i ) for i, which yields an exact sequence N i X i N i+1. ( i ) Each N i is an object in cores X. By induction on i, Lemma 1.6 (a) implies that N i W. We will construct exact sequences N W N 1, ( ) N 1 N 1 N 1, ( ) such that ( )ishom A (, W)-exact, W is an object in W, and N 1 is an object in cores W such that N 1 W. Inducting on i, this will yield exact sequences N i W i N i+1, ( i ) N i+1 N i+1 N i+1, ( i ) such that ( i ) is Hom A (, W)-exact, W i is an object in W, and N i+1 is an object in cores W such that N i+1 W. Splicing together the sequences ( i) will then yield a proper W-coresolution of N. Consider the following pushout diagram, with the given exact sequence as the top row and with ( ) as the leftmost column. N N N X V N (1) N 1 N 1

6 486 S. SATHER-WAGSTAFF, T. SHARIF AND D. WHITE Since N W and X W, the middle row of (1) ishom A (, W)-exact and V W by Lemma 1.6 (a). Note that X is in cores W by Lemma 1.8. Since N is in cores W, Lemma 1.9(a) implies that V is in cores W. Hence, a proper W-coresolution of V provides an exact sequence V W N 1 that is Hom A (, W)-exact with objects W in W and N 1 in cores W. By assumption, we have X Wand W X, and so W W because W is in W. Since V W, Lemma 1.6(a) implies that N 1 W. With the center column of (1) this yields another pushout diagram, and we shall show that the middle row and the rightmost column satisfy the conditions for ( ) and ( ), respectively. N V N 1 N W N 1 (2) N 1 N 1 We have already seen that W is in W and N 1 is in cores W. Since N 1 Wand N 1 W, the rightmost column of (2) with Lemma 1.6(a) yields N 1 W, and so the middle row of (2) is Hom A (, W)-exact. Next is a key lemma for both Theorems A and B from the introduction. Lemma 2.2. Assume that W is an injective cogenerator for X, and that V is a projective generator for Y. (a) If M is an object in cores X and M W, then M is in cores W. (b) If N is an object in res Ỹ and V N, then N is in res Ṽ. Proof. We prove part (a); the proof of part (b) is dual. Let M X be a proper X -coresolution of M. Setting M = Im( X ) yields an exact sequence M X M (3) that is Hom A (, X )-exact. Since W is a cogenerator for X there is an exact sequence with objects W in W and X in X. X W X (4)

7 STABILITY OF GORENSTEIN CATEGORIES 487 Consider the pushout diagram with (3) as the top row and with (4) as the middle column. M X M M W U (5) X X We will show that U is an object in cores W and that the middle row of (5) is Hom A (, W)- exact. It will then follow that a proper W-coresolution for M can be obtained by splicing the middle row of (5) with a proper W-coresolution of U. The object M is in cores X by construction, and X and X are in X. Thus, X is an object in cores W by Lemma 1.8, and X W and X W by hypothesis. Since the top row of (5) ishom A (, X )-exact, the assumption W X implies that the top row of (5) is Hom A (, W)-exact. Hence, the assumption that M W yields M W by Lemma 1.6(a). With the rightmost column of (5), Lemma 2.1(a) implies that U is an object in cores W. Since X W and M W, Lemma 1.6(a) yields U W, and so the middle row of (5) is Hom A (, W)-exact. The last result in this section is a tool for Proposition 4.6. Lemma 2.3. For n =, 1, 2,...,t, let X n and Y n be subcategories of A. (a) Assume that X n is a cogenerator for X n+1 for each n and X t X.IfX t is closed under extensions, then X is an injective cogenerator for X t. (b) Assume that Y n is a generator for Y n+1 for each n and Y Y t.ify t is closed under extensions, then Y is a projective generator for Y t. Proof. We prove part (a); the proof of part (b) is dual. Since X t X by assumption, it remains to show that X is a cogenerator for X t. Fix an object X t in X t. By reverse induction on i<t, we will construct exact sequences X t X i X (i) t ( i ) with objects X i in X i and X (i) t in X t. Since X t 1 is a cogenerator for X t, the sequence ( t 1 )is known to exist. By induction, we assume that ( i ) has been constructed and construct ( i 1 ) from it. From ( i ) we have the object X i in X i. Since X i 1 is a cogenerator for X i, there is an exact sequence with X i 1 in X i 1 and X i in X i. X i X i 1 X i ( i )

8 488 S. SATHER-WAGSTAFF, T. SHARIF AND D. WHITE Consider the pushout diagram X t Xt Xi X i 1 X i (i) X t X (i 1) t X i with ( i ) as the leftmost column and ( i ) as the middle row. The object X i is in X i, and hence in X t. Since X (i) t is also in X t, the exactness of the bottom row, with the fact that X t is closed under extensions, implies that X (i 1) t is in X t, so the center column of the diagram is the desired sequence ( i 1 ). 3. Categories of interest Much of the motivation for this work comes from module categories. In reading this paper, the reader may find it helpful to keep in mind the examples outlined in the next few paragraphs, wherein R is a commutative ring. Definition 3.1. Let M(R) denote the category of R-modules. To be clear, we write P(R) for the subcategory of projective R-modules and I(R) for the subcategory of injective R-modules. If X (R) is a subcategory of M(R), then X f (R) is the subcategory of finitely generated modules in X (R). Also set Ab = M(Z), the category of abelian groups. The study of semidualizing modules, defined next, was initiated independently (with different names) by Foxby [1], Golod [13], and Vasconcelos [19]. Definition 3.2. An R-module C is semidualizing if it satisfies the following. (1) The R-module C admits a (possibly unbounded) resolution by finite rank free R-modules. (2) The natural homothety map R Hom R (C, C) is an isomorphism. (3) Ext 1 R (C, C) =. A finitely generated projective R-module of rank 1 is semidualizing. If R is Cohen Macaulay, then C is dualizing if it is semidualizing and id R (C) is finite. Based on the work of Enochs and Jenda [8], the following notions were introduced and studied in this generality by Holm and Jørgensen [15] and White [2].

9 STABILITY OF GORENSTEIN CATEGORIES 489 Definition 3.3. Let C be a semidualizing R-module, and set P C (R) = the subcategory of modules P R C where P is R-projective I C (R) = the subcategory of modules Hom R (C, I) where I is R-injective. Modules in P C (R) and I C (R) are called C-projective and C-injective, respectively. A complete PP C -resolution is a complex X of R-modules satisfying the following. (1) The complex X is exact and Hom R (, P C (R))-exact. (2) The R-module X i is projective if i and X i is C-projective if i<. An R-module M is G C -projective if there exists a complete PP C -resolution X such that M Coker( 1 X ), in which case X is a complete PP C -resolution of M. Weset GP C (R) = the subcategory of G C -projective R-modules. Projective R-modules and C-projective R-modules are G C -projective, and P C (R) is an injective cogenerator for GP C (R) by[15, Lemma 2.5 and Proposition 2.13] and [2, Propositions 3.2 and 3.9]. A complete I C I-coresolution is a complex Y of R-modules such that: (1) The complex Y is exact and Hom R (I C (R), )-exact. (2) The R-module Y i is injective if i and Y i is C-injective if i>. An R-module N is G C -injective if there exists a complete I C I-coresolution Y such that N Ker( Y ), in which case Y is a complete I C I-coresolution of N. Weset GI C (R) = the subcategory of G C -injective R-modules. An R-module that is injective or C-injective is G C -injective, and I C (R) is a projective generator for GI C (R) by[15, Lemma 2.6, and Proposition 2.13] and results dual to [2, Propositions 3.2 and 3.9]. The next definition was first introduced by Auslander and Bridger [1, 2] in the case C = R, and in this generality by Golod [13] and Vasconcelos [19]. Definition 3.4. Assume that R is noetherian and C is a semidualizing R-module. A finitely generated R-module H is totally C-reflexive if: (1) Ext 1 R (H, C) ==Ext1 R (Hom R(H, C),C), and (2) the natural biduality map H Hom R (Hom R (H, C),C) is an isomorphism. Each finitely generated R-module that is either projective or C-projective is totally C-reflexive. We set G C (R) = the subcategory of totally C-reflexive R-modules and G(R) =G R (R). The equality G C (R) =GP f C(R) holds by [2, Theorem 5.4], and PC f (R) is an injective cogenerator for G C (R) by[2, Proposition 3.9, Lemma 5.3 and Theorem 5.4]. Over a noetherian ring, the next categories were introduced by Avramov and Foxby [4] when C is dualizing, and by Christensen [7] for arbitrary C. (Note that these works (and others) use the notation A C (R) and B C (R) for certain categories of complexes, while our categories consist precisely of the modules in these other categories.) In the non-noetherian setting, these definitions are taken from [16, 2]. Definition 3.5. Let C be a semidualizing R-module. The Auslander class of C is the subcategory A C (R) ofr-modules M such that: (1) Tor R 1(C, M) ==Ext 1 R (C, C R M), and (2) the natural map M Hom R (C, C R M) is an isomorphism.

10 49 S. SATHER-WAGSTAFF, T. SHARIF AND D. WHITE The Bass class of C is the subcategory B C (R) ofr-modules N such that: (1) Ext 1 R (C, M) ==TorR 1(C, Hom R (C, M)), and (2) the natural evaluation map C R Hom R (C, N) N is an isomorphism. For a discussion of the next subcategory, consult [9, Section 5.3]. Definition 3.6. The category of flat cotorsion R-modules is the subcategory F (R) offlat R-modules F such that Ext 1 R (F,F) = for each flat R-module F. Gerko [12] introduced our final subcategory of interest. Definition 3.7. Assume that (R, m,k) is local and noetherian. The complexity of a finitely generated R-module M is cx R (M) = inf{d N there exists a c> such that β R n (M) cn d 1 for n }, where β R n (M) = rank k (Tor R i (M,k)) is the nth Betti number of M. Let G (R) denote the subcategory of modules in G(R) with finite complexity. 4. Gorenstein subcategories In this section, we introduce and study the Gorenstein subcategory G(X ). Definition 4.1. An exact complex in X is totally X -acyclic if it is Hom A (X, )-exact and Hom A (, X )-exact. Let G(X ) denote the subcategory of A with objects of the form M Coker( 1 X ) for some totally X -acyclic complex X; we say that X is a complete X -resolution of M. Note that the isomorphisms Hom A (X X,X) Hom A (X,X) Hom A (X,X), Hom A (X, X X ) Hom A (X, X ) Hom A (X, X ) show that the direct sum of totally X -acyclic X -complexes is totally X -acyclic, and hence G(X ) is additive. Set G (X )=X and G 1 (X )=G(X ), and inductively set G n+1 (X )=G(G n (X )) for n 1. Remark 4.2. Any contractible X -complex is totally X -acyclic; see Definition 1.3. In particular, for any object X in X, the complex X id X X is a complete X -resolution, and so X is an object in G(X ). Hence X G(X ), and inductively G n (X ) G n+1 (X ) for each n. There is a containment G(X ) res X cores X. Indeed, If M is an object in G(X ) with complete X -resolution X, then the hard truncation X is a proper X -resolution of M and X < is a proper X -coresolution of M. The orthogonality properties documented next will be very useful in the sequel; compare to [6, Proposition 4.2.5].

11 STABILITY OF GORENSTEIN CATEGORIES 491 Proposition 4.3. If X Wand V Y, then G n (X ) res Ŵ and cores V G n (Y) for each n 1. In particular, if W W, then G n (W) res Ŵ and cores Ŵ Gn (W) for each n 1. Proof. Assuming that X W, we will show that G(X ) W; the conclusion that G n (X ) W will follow by induction, and G n (X ) res Ŵ will then follow from Lemma 1.7. The other conclusion is verified dually. Let M be an object in G(X ) with complete X -resolution X, and let W be an object in W. For each integer i set M i = Coker( i X). Note that M M 1. The exact sequence ɛ M i i+1 Xi 1 M i is Hom A (, X )-exact, and so it is Hom A (,W)-exact. In particular, the map Hom A (ɛ i,w) is surjective. Since X W, part of the beginning of the associated long exact sequence in Ext A (,W)is Hom A (X i 1,W) Hom A(ɛ i,w ) Hom A (M i+1,w) Ext 1 A(M i,w) so the surjectivity of Hom A (ɛ i,w) implies that Ext 1 A(M i,w) =. In particular, Ext 1 A(M,W) Ext 1 A(M 1,W)=. If j 2, then the remainder of the long exact sequence yields isomorphisms Ext j A (M i,w) Ext j 1 A (M i+1,w). Inductively, this yields the second isomorphism in the next sequence and the desired vanishing Ext j A (M,W) Ext j A (M 1,W) Ext 1 A(M j,w)=. We next present a horseshoe lemma for complete X -resolutions; compare to [6, Corollary 4.3.5(a)]. Proposition 4.4. Consider an exact sequence in A M M M that is Hom A (X, )-exact and Hom A (, X )-exact. If M and M are objects in G(X), then so is M. Furthermore, given complete X -resolutions X and X of M and M, respectively, there is a degreewise split exact sequence of complexes X X X such that X is a complete X -resolution of M, the induced sequence Coker( 1 X ) Coker( 1 X ) Coker( 1 X ) is equivalent to the original sequence, and ( ) n X X n f = n n X for each n Z. Proof. Let X and X be complete X -resolutions for M and M, respectively. Lemma 1.9 yields a degreewise split exact sequence of complexes X X X such that M Coker( 1 X ). Note that each X i X i X i is in X. Since the complexes X and X are both Hom A (X, )-exact and Hom A (X, )-exact, the same is true of X. Therefore, X is a complete X -resolution of M.

12 492 S. SATHER-WAGSTAFF, T. SHARIF AND D. WHITE Corollary 4.5. If W W, then G(W) is closed under extensions. Proof. Proposition 4.3 implies that G(W) Wand W G(W). Hence, any exact sequence M M M with M and M objects in G(W) is Hom A (W, )-exact and Hom A (, W)-exact. Now apply Proposition 4.4. It is unclear in general whether W is an injective cogenerator for G n (X ) without the extra hypotheses in our next result. Proposition 4.6. Fix an integer n 1. (a) If W is an injective cogenerator for X and G n (X ) is closed under extensions, then W is an injective cogenerator for G n (X ). (b) If V is a projective generator for Y and G n (Y) is closed under extensions, then V is a projective generator for G n (Y). Proof. We prove part (a); the proof of part (b) is dual. Set X = W, X 1 = X, and X t = G t 1 (X ) for t =2,...,n+ 1. By Proposition 4.3 we know that G n (X ) W, so the desired conclusion follows from Lemma 2.3. In Section 5 we document the consequences of the following result for the examples of Section 3. Corollary 4.7. generator for G(W). If W W, then W is both an injective cogenerator and a projective Proof. This follows from Corollary 4.5 and Proposition 4.6. The next result extends part of Remark 4.2 and represents a first step in the proof of Theorem A from the introduction. Theorem 4.8. Assume that W is an injective cogenerator for X and that V is a projective generator for Y. (a) If X is closed under extensions, then G n (X ) cores W for each n. (b) If Y is closed under extensions, then G n (Y) res Ṽ for each n. Proof. We prove part (a) by induction on n; the proof of part (b) is dual. The case n =is given in Lemma 1.8. When n = 1, note that an object M in G(X ) is in cores X by Remark 4.2, and one has M W by Proposition 4.3; now apply Lemma 2.2. Assume that n>1 and G n 1 (X ) cores W. Fix an object M in G n (X ) and a complete G n 1 (X )-resolution G of M. By definition, the complex G is exact and there is an isomorphism M Ker( 1). G For each integer j, set M j = Ker( j G) and observe that each M j is an object in G n (X ). Since each object G j is in G n 1 (X ), Proposition 4.3 implies that M j Wand G j W for each integer j, and we consider the exact sequences M j G j M j 1. ( j ) Our induction assumption implies that each object G j is in cores W.

13 STABILITY OF GORENSTEIN CATEGORIES 493 By induction on i, we construct exact sequences in A M W W 1 W i U i ( i ) M (i+2) U i V i ( i ) U i W (i+1) U (i+1) ( i ) M (i+3) U (i+1) V (i+1) ( i ) satisfying the following properties: (a i ) the objects W,...,W (i+1) are in W; (b i ) the sequence ( i )ishom A (, W)-exact; (c i ) one has U i W and U (i+1) W; (d i ) one has V i W and V (i+1) W; (e i ) one has V i and V (i+1) in cores W. The sequence ( i ) is obtained by splicing the sequences ( ), ( ),..., ( i 1 ). Continuing to splice inductively, conditions (a i ) (c i ) show that this process yields a proper W-coresolution of M, as desired. We begin with the base case i =. The membership G 1 cores W yields a proper W-coresolution of G 1 and hence an exact sequence G 1 W V (6) that is Hom A (, W)-exact and with objects W W and V cores W. Using the conditions G 1 W and W W, Lemma 1.6(a) implies that V W. Consider the pushout diagram with ( 1 ) as the top row and (6) as the middle column. M G 1 M 2 M W U (7) V V Applying Lemma 1.6(a) to the rightmost column of this diagram, the conditions M 2 W and V Wimply that U W. For each object W W, use the condition U W with the long exact sequence in Ext A (,W ) asociated to the middle row of this diagram to conclude that this row is Hom A (, W)-exact. Set ( ) equal to the middle row of (7), and set ( ) equal to the rightmost column of (7). Construct the next pushout diagram using ( 2 ) in the top

14 494 S. SATHER-WAGSTAFF, T. SHARIF AND D. WHITE row and the rightmost column of (7) in the left column. M 2 G 2 M 3 U Z M 3 (8) V V As in the above discussion, the condition V W implies that the middle column of (8) is Hom A (, W)-exact. Using Lemma 1.6(a) with this column, the conditions G 2 Wand V W yield Z W. We know that G 2 and V are in cores W, so an application of Lemma 1.9(b) to this column implies that Z cores W. A proper W-coresolution of Z provides an exact sequence Z W 1 V 1 (9) that is Hom A (, W)-exact and with objects W 1 W and V 1 cores W. Again using Lemma 1.6(a), the conditions Z Wand W 1 Wconspire with the Hom A (, W)-exactness of (9) to imply that V 1 W. Consider the next pushout diagram, with the middle row of (8) as the top row and with (9) as the middle column. U Z M 3 U W 1 U 1 (1) V 1 V 1 Set ( ) equal to the middle row of this diagram, and ( ) equal to the rightmost column. Due to Lemma 1.6(a), the conditions M 3 W and V 1 W imply that U 1 W. Thus, conditions (a ) (e ) are satisfied, establishing the base case. For the induction step, assume that the exact sequences ( i ), ( i ), ( i ), and ( i ) have been constructed satisfying conditions (a i ) (e i ). Note that condition (c i ) implies that the sequence ( i )ishom A (, W)-exact. Thus, we may splice together the sequences ( i ) and ( i ) to construct the sequence ( i+1 ) which is exact and Hom A (, W)-exact and such that W,...,W (i+1) W. Also, set ( i+1 )=( i ).

15 STABILITY OF GORENSTEIN CATEGORIES 495 The next pushout diagram has ( i+3 ) in the top row and ( i ) in the left column. M (i+3) G (i+3) M (i+4) U (i+1) Z (i+1) M (i+4) (11) V = (i+1) V (i+1) With the long exact sequence in Ext A (, ), the condition V (i+1) W implies that the middle column of (11) is Hom A (, W)-exact. Using Lemma 1.6(a) with this column, the conditions G (i+3) W and V (i+1) W yield Z (i+1) W.AsG (i+3) and V (i+1) are in cores W, apply Lemma 1.9(b) to this column to conclude that Z (i+1) cores W. A proper W-coresolution of Z (i+1) provides an exact sequence Z (i+1) W (i+2) V (i+2) (12) that is Hom A (, W)-exact and with objects W (i+2) W and V (i+2) cores W. Again using Lemma 1.6(a) and the Hom A (, W)-exactness of (12), the conditions Z (i+1) W and W (i+2) W imply that V (i+2) W. Consider the next pushout diagram, with the middle row of (11) as the top row and with (12) as the middle column. U (i+1) Z (i+1) M (i+4) U (i+1) W (i+2) U (i+2) (13) V = (i+2) V (i+2) Set ( i+1 ) equal to the middle row of this diagram, and set ( i+1 ) equal to the rightmost column. With Lemma 1.6(a), the conditions M (i+4) W and V (i+2) W imply that U (i+2) W. Thus, conditions (a i+1 ) (e i+1 ) are satisfied, establishing the induction step. What follows is the second step in the proof of Theorem A from the introduction. See Example 5.9 for the necessity of the cogeneration hypothesis.

16 496 S. SATHER-WAGSTAFF, T. SHARIF AND D. WHITE Theorem 4.9. If X is closed under extensions and W is both an injective cogenerator and a projective generator for X, then G n (X ) G(W) for each n 1. Proof. Let N be an object in G n (X ). By Theorem 4.8 we know that N admits a proper W-resolution W N and a proper W-coresolution N W. We will show that + W is Hom A (, W)-exact and that + W is Hom A (W, )-exact. Since we already know that + W is Hom A (W, )-exact and that + W is Hom A (, W)-exact, this will show that the concatenated complex W 1 W W W 1 is a complete W-resolution of N, completing the proof. We will show that + W is Hom A (, W)-exact; the proof of the other fact is dual. For each i, there is an exact sequence N i+1 W i N i. ( i ) We have N = N and so N W is true by Proposition 4.3, and W i W by assumption. Using Lemma 1.6(a), an induction argument implies that N i W for each i. It follows that ( i )ishom A (, W)-exact, and it follows that + W is Hom A (, W)-exact, as desired. Theorem A from the introduction follows from the next result; see Example 5.3. Corollary 4.1. If W W, then G n (W) =G(W) for each n 1. Proof. Note that Corollaries 4.5 and 4.7 imply that G(W) is closed under extensions and that W is both an injective cogenerator and a projective generator for G(W). We argue by induction on n, the case n = 1 being trivial. For n>1, if G n 1 (W) =G(W), then setting X = G(W) in Theorem 4.9 yields the final containment in the next sequence and hence the desired conclusion. G(W) G n (W) =G(G n 1 (W)) = G(G(W)) G(W), With Corollary 4.5, the final two results of this section contain Theorem B from the introduction; compare to [6, Corollary 4.3.5]. Proposition If W W, then G(W) is closed under direct summands. Proof. Let A and A be objects in A such that A A is in G(W). We construct proper W-resolutions W A and W A such that (W ) + and (W ) + are Hom A (, W)-exact. Dually, one constructs proper W-coresolutions A V and A V such that + V and + V are Hom A (W, )-exact, and this shows that A and A are in G(W). Observe first that A and A both admit (augmented) proper G(W)-resolutions X = ( id ) A A ( id ) A A ( id ) A, X = ( id ) A A ( id ) A A ( id) A,

17 STABILITY OF GORENSTEIN CATEGORIES 497 where properness follows from the contractibility of X and X. From Proposition 4.3 we know that W (A A ), so the additivity of Ext A implies that W A and W A. Lemma 2.2(b) and Corollary 4.7 imply that A and A admit proper W-resolutions W A and W A because W G(W) and A,A res G(W) by the preceeding argument. Hence, W W A A is a proper W-resolution. We show that (W ) + and (W ) + are Hom A (, W)-exact. As A A is in G(W), it admits a proper W-resolution W A A such that W + is Hom A (, W)-exact. Hence, the resolutions W and W W are homotopy equivalent. Because W + is Hom A (, W)-exact, we know that (W W ) + is also Hom A (, W)-exact, and so (W ) + and (W ) + are Hom A (, W)-exact. Theorem Assume that W W. (a) If W is closed under kernels of epimorphisms, then so is G(W). (b) If W is closed under cokernels of monomorphisms, then so is G(W). Proof. We prove part (a); the proof of part (b) is dual. Consider an exact sequence in A with objects N and N in G(W): N N τ N. (14) Let W and W be complete W-resolutions of N and N, respectively. We first construct a commutative diagram of the following form. W 2 W 1 W1 W τ 1 τ W 2 W 1 W 1 W π N τ π N ɛ W 1 W 1 τ 1 ɛ W 1 W 2 W 2 τ 2 W 1 W 2 Here ɛπ = W and ɛ π = W. Since (W ) + is a chain complex and (W )+ is Hom A (W, ) exact, one can successively lift τ to the left as in the diagram; argue as in [14, Proposition 1.8]. Dually, since + W < is a chain complex and + W < is Hom A (, W) exact, one can successively lift τ to the right as in the diagram. Thus, we have constructed a morphism of chain complexes τ : W W such that the induced map Coker( 1 W ) Coker( W 1 ) is equivalent to τ. Next, we show that there exist a complex W and a morphism τ : W W W satisfying the following properties. (i) The complex W is contractible and W n is in W for each n Z; (ii) The morphism τ n is an epimorphism for each n Z; (iii) The natural monomorphism W ɛ W W satisfies τ = τ ɛ. The complex W = Σ 1 Cone(id W ) is contractible; see Definition 1.3. Let f : W W denote the composition of the natural morphisms W = Σ 1 Cone(id W ) W. Note that each f n is a split epimorphism. It follows that the homomorphisms τ n =( τ n f n ): W n W n W n describe a morphism of complexes satisfying the desired properties. Because of property (i) the complex W is a complete W-resolution; see Remark 4.2. Set Ñ = Coker( W 1 ), which is an object in G(W) with complete resolution W. Furthermore, one has Coker( W W 1 ) N Ñ, W 2 (15)

18 498 S. SATHER-WAGSTAFF, T. SHARIF AND D. WHITE and the morphism τ induces a homomorphism f=(τ π) N Ñ N. Because τ is surjective, the map f is also surjective. We will show that Ker(f) ising(w), and then we will show that N = Ker(τ) ising(w). The morphism τ is degreewise surjective. As W is closed under kernels of epimorphisms, it follows that the complex W = Ker( τ) consists of objects in W. The next exact sequence shows that W is exact because W, W, and W are so: This sequence induces a second exact sequence W W W τ W. W W W W with associated long exact sequence of the form Coker( W 1 ) N f N. Thus, we have Ker(f) Coker( 1 W ). To show that Ker(f) ising(w), it suffices to show that W is Hom A (W, )-exact and Hom A (, W)-exact. For each object U in W, the next sequence of complexes is exact as W W: Hom A (U, W ) Hom A (U, W ) Hom A (U, W ). Since W and W are Hom A (W, )-exact, the associated long exact sequence shows that W is also Hom A (W, )-exact. Dually, one shows that W is Hom A (, W)-exact, thus showing that Ker(f) ising(w). To see that N is in G(W), consider the following pullback diagram, with (14) as the rightmost column and with the middle row that is the natural split exact sequence. γ Ñ Ker(f) α β Ñ N Ñ δ N N (16) f N τ N Let σ : N Ñ Ñ denote the natural surjection. It follows that σβ =id. Since α is an Ñ isomorphism, the equality δγ = βα implies that (α 1 σδ)γ = α 1 σβα = α 1 α =idñ, and so the top row of (16) is split exact. Hence, the object N is a direct summand of Ker(f). We have shown that Ker(f) is in G(W). The category G(W) is closed under direct summands by Proposition 4.11, and so N is in G(W) as desired.

19 STABILITY OF GORENSTEIN CATEGORIES Consequences for categories of interest Let R be a commutative ring, and let C be a semidualizing R-module. We now apply the results of Section 4 to the examples in Section 3. We begin with some computations. Example 5.1. The relevant definitions yield equalities G(P(R)) = GP(R) and G(I(R)) = GI(R). If R is noetherian, then G(P f (R)) = G(R). The next result generalizes the previous example. Proposition 5.2. Let R be a commutative ring. If C is R-semidualizing, then G(P C (R)) = GP C (R) B C (R) and G(I C (R)) = GI C (R) A C (R). If, further, R is noetherian, then G(PC f (R)) = G C(R) B C (R). Proof. We will prove the first equality; the others are proved similarly. For one containment, let M be an object in G(P C (R)). To show that M is an object in GP C (R), we use [2, Proposition 3.2]: it suffices to show that M admits a proper P C (R)-coresolution and M P C (R). The first of these is given in Remark 4.2 which says that M is in cores P C (R); the second one is given in Proposition 4.3, which implies that G(P C ) P C. To show that M is an object in B C (R), we need to verify Ext 1 R (C, M) = and TorR 1(C, Hom R (C, M))= and M C R Hom R (C, M). The first of these is given in Proposition 4.3, which implies that P C G(P C ), and the others are in [18, Proposition 2.2(a)]. For the reverse containment, fix an object N in GP C (R) B C (R). Since N is in GP C (R), it admits a complete PP C -resolution Y, so the complex Y < is a proper P C (R)-coresolution of N. Also, N admits a proper P C (R)-resolution Z by [18, Corollary 2.4(a)] as N is in B C (R). Once it is shown that Y < + is Hom R(P C (R), )-exact and Z + is Hom R (, P C (R))-exact, a complete PP C -resolution of N will be obtained by splicing Z and Y <. To see that Y < + is Hom R(P C (R), )-exact, set N () = N and N (i) = Coker( i 1 Y ) for each i 1. From [16, Lemma 5.1(a)], we know that Y i is in B C (R) for each i 1. Since N is also in B C (R), an induction argument using the exact sequence N (i 1) Y i N (i) ( i ) implies that N (i) is in B C (R) for each i 1. For each projective R-module P, this yields the vanishing in the next sequence Ext 1 R(P R C, N (i) ) Hom R (P, Ext 1 R(C, N (i) ))=, while the isomorphism is from Hom tensor adjointness. It follows that each sequence ( i )is Hom R (P C (R), )-exact, and thus so is Y < +. To see that Z + is Hom R (, P C (R))-exact, it suffices to let P be projective and to justify the following sequence for i 1: H i (Hom R (Z +,P R C))=H i (Hom R (Z, P R C)) Ext i R(N,P R C)=. The isomorphism is derived from [18, Corollary 4.2(a)] because N and P R C are in B C (R). The vanishing follows because N is in GP C (R) and GP C (R) P C (R); see [2, Proposition 3.2]. We now outline the consequences of Corollaries 4.7 and 4.1 for the examples of Section 3. The first example below contains Theorem A from the introduction.

20 5 S. SATHER-WAGSTAFF, T. SHARIF AND D. WHITE Example 5.3. The category P C (R) is an injective cogenerator and a projective generator for GP C (R) B C (R), and G n (P C (R)) = GP C (R) B C (R) for each n 1. Hence, P(R) isan injective cogenerator and a projective generator for GP(R), and G n (P(R)) = GP(R). If A has enough projectives, then P(A) is an injective cogenerator and a projective generator for G(P(A)), and G n (P(A)) = G(P(A)). Example 5.4. The category I C (R) is an injective cogenerator and a projective generator for GI C (R) A C (R), and G n (I C (R)) = GI C (R) A C (R) for each n 1. Hence, I(R) isan injective cogenerator and a projective generator for GI(R), and G n (I(R)) = GI(R). If A has enough injectives, then I(A) is an injective cogenerator and a projective generator for G(I(A)), and G n (I(A)) = G(I(A)). Example 5.5. Assume that R is noetherian. Then PC f (R) is an injective cogenerator and a projective generator for G C (R) B C (R), and G n (PC f (R)) = G C(R) B C (R). In particular, P f (R) is an injective cogenerator and a projective generator for G(R), and G n (P f (R)) = G(R). Example 5.6. The category F (R) is an injective cogenerator and a projective generator for G(F (R)), and G n (F (R)) = G(F (R)) for each n 1. With Proposition 4.3 and Corollary 4.5 in mind, we now show that W Wneed not imply that G(W) G(W). Example 5.7. Let (R, m,k) be a local, nonregular, Gorenstein, artinian ring. With W = P f (R), we have G(W) =G(R) =M f (R) where the last equality holds because R is artinian and Gorenstein; see [6, Theorems and 1.4.9]. We know that Ext 1 R (k, k) since R is nonregular, and so G(W) G(W). We conclude with some questions and final observations. Questions 5.8. Must there be an equality G n (X )=G(X) for each n 1? Is G(X ) always exact? Is G(X ) always closed under kernels of epimorphisms or cokernels of monomorphisms? Must G(W) be contained in G(X )? Can G(F(R)) or G(F (R)) or G(G (R)) be identified as in Proposition 5.2? The final examples are presented with an eye toward the last question of Questions 5.8. Example 5.9. If (R, m) is a noetherian local ring and dim(r) 1, then G(F(R)) G(P(R)). Indeed, the ring of formal power series R[[X]] is a flat R-module, so it is in G(F(R)). Suppose by way of contradiction that R[[X]] is in G(P(R)). First note that [17, Proposition 6] yields pd R (R[[X]]) <, and so [14, Proposition 2.7] implies that pd R (R[[X]]) = G-pd R (R[[X]]) =. It follows that R[[X]] is projective, contradicting [5, Theorem 2.1]. From this it follows that the conclusion of Theorem 4.9 need not hold if W is not a cogenerator for X. To see this, assume that R is m-adically complete. Standard results combine to show that P(R) is a projective generator for F(R) and that F(R) is closed under extensions. Furthermore, one has F(R) P(R) by[9, Theorem ]. With Theorem 4.9, the previous example provides the next result.

21 STABILITY OF GORENSTEIN CATEGORIES 51 Corollary 5.1. If R is a complete local notherian ring and dim(r) 1, then P(R) is not a cogenerator for F(R). Example Let (R, m) be a noetherian local ring. If R is not m-adically complete, then G(F (R)) G(P(R)). The m-adic completion R is flat and cotorsion; see, for example, [9, Theorem ]. Arguing as in Example 5.9, it then suffices to note that R is not projective by [11, Theorem A]. Example Assume that R is local and noetherian. Using Example 5.3 and [12, Proposition 2.8], it is straightforward to show that P f (R) is an injective cogenerator and a projective generator for G (R) and that G (R) is closed under extensions. Theorem 4.9 now implies that G n (G (R)) G(R) for each n 1. Acknowledgements. We are indebted to the referee for the careful reading of this work. References 1. M. Auslander, Anneaux de Gorenstein, et torsion en algèbre commutative, Séminaire d Algèbre Commutative dirigé par Pierre Samuel, 1966/67 (Secrétariat Mathématique, Paris, 1967) 69; MR 37 # M. Auslander and M. Bridger, Stable module theory, Memoirs of the American Mathematical Society 94 (American Mathematical Society, Providence, RI, 1969); MR 42 # M. Auslander and R.-O. Buchweitz, The homological theory of maximal Cohen Macaulay approximations, Colloque en l honneur de Pierre Samuel, Orsay, 1987, Mém. Soc. Math. France (N.S.) 38 (1989) 5 37; MR (91h:131). 4. L. L. Avramov and H.-B. Foxby, Ring homomorphisms and finite Gorenstein dimension, Proc. London Math. Soc. (3) 75 (1997) ; MR 98d: R.-O. Buchweitz and H. Flenner, Power series rings and projectivity, Manuscripta Math. 119 (26) ; MR (27a:1325). 6. L. W. Christensen, Gorenstein dimensions, Lecture Notes in Mathematics 1747 (Springer, Berlin, 2); MR 22e: L. W. Christensen, Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc. 353 (21) ; MR 22a: E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 22 (1995) ; MR (97c:1611). 9. E. E. Enochs and O. M. G. Jenda, Relative homological algebra, de Gruyter Expositions in Mathematics 3 (de Gruyter, Berlin, 2); MR (21h:1613). 1. H.-B. Foxby, Gorenstein modules and related modules, Math. Scand. 31 (1972) (1973); MR 48 # A. Frankild and S. Sather-Wagstaff, Detecting completeness from Ext-vanishing, Preprint, 26, arxiv:math.ac/66736, Proc. Amer. Math. Soc., to appear. 12. A. A. Gerko, On homological dimensions, Mat. Sb. 192 (21) (Russian), Sb. Math. 192 (21) (English); MR 22h: E. S. Golod, G-dimension and generalized perfect ideals, Algebraic geometry and its applications, Tr. Mat. Inst. Steklova 165 (1984) 62 66; MR 85m: H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (24) ; MR (24k:1613). 15. H. Holm and P. Jørgensen, Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 25 (26) ; MR H. Holm and D. White, Foxby equivalence over associative rings, Preprint, 26, arxiv:math.ac/611838, J. Math. Kyoto Univ., to appear. 17. C. U. Jensen, On the vanishing of lim (i), J. Algebra 15 (197) ; MR (41 #546). 18. R. Takahashi and D. White, Homological aspects of semidualizing modules, Preprint, 27, arxiv:math.ac/ W. V. Vasconcelos, Divisor theory in module categories, North-Holland Mathematics Studies 14, Notas de Matemática 53 [Notes on Mathematics 53] (North-Holland, Amsterdam, 1974); MR (58 #16637). 2. D. White, Gorenstein projective dimension with respect to a semidualizing module, Preprint, 26, arxiv:math.ac/611711, J. Commutative Algebra, to appear.

22 52 STABILITY OF GORENSTEIN CATEGORIES Sean Sather-Wagstaff Department of Mathematical Sciences Kent State University Mathematics and Computer Science Building Summit Street Kent OH USA Current address: Department of Mathematics 3 Minard Hall North Dakota State University Fargo, ND USA Sean Sather-Wagstaff@ndsu edu ssatherw/ Tirdad Sharif School of Mathematics Institute for Studies in Theoretical Physics and Mathematics PO Box Tehran Iran sharif@ipm ir personalinfo.jsp?peoplecode=ip46 Diana White Department of Mathematics University of Nebraska 23 Avery Hall Lincoln, NE USA Current address: Department of Mathematics LeConte College 1523 Greene Street University of South Carolina Columbia, SC 2928 USA dwhite@math sc edu dwhite/