AB-Contexts and Stability for Gorenstein Flat Modules with Respect to Semidualizing Modules

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1 Algebr Represent Theor (2011) 14: DOI /s AB-ontexts and Stability for Gorenstein Flat Modules with Respect to Semidualizing Modules Sean Sather-Wagstaff Tirdad Sharif Diana White Received: 3 January 2009 / Accepted: 14 July 2009 / Published online: 23 December 2009 Springer Science+Business Media B.V Abstract We investigate the properties of categories of G -flat R-modules where is a semidualizing module over a commutative noetherian ring R. We prove that the category of all G -flat R-modules is part of a weak AB-context, in the terminology of Hashimoto. In particular, this allows us to deduce the existence of certain Auslander- Buchweitz approximations for R-modules of finite G -flat dimension. We also prove that two procedures for building R-modules from complete resolutions by certain subcategories of G -flat R-modules yield only the modules in the original subcategories. Presented by Juergen Herzog. TS is supported by a grant from IPM, (No ). S. Sather-Wagstaff (B) Department of Mathematics, North Dakota State University, Dept 2750, PO Box 6050, Fargo, ND , USA Sean.Sather-Wagstaff@ndsu.edu URL: ssatherw/ T. Sharif School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box , Tehran, Iran sharif@ipm.ir URL: D. White Department of Mathematical & Statistical Sciences, University of olorado Denver, ampus Box 170, P.O. Box , Denver, O , USA Diana.White@ucdenver.edu URL: diwhite/

2 404 S. Sather-Wagstaff et al. Keywords AB-contexts Auslander-Buchweitz approximations Auslander classes Bass classes otorsion Gorenstein flats Gorenstein injectives Semidualizing Mathematics Subject lassifications (2000) D02 13D05 1 Introduction Auslander and Bridger [1, 2] introduce the modules of finite G-dimension over a commutative noetherian ring R, in part, to identify a class of finitely generated R- modules with particularly nice duality properties with respect to R. They are exactly the R-modules which admit a finite resolution by modules of G-dimension 0. As a special case, the duality theory for these modules recovers the well-known duality theory for finitely generated modules over a Gorenstein ring. This notion has been extended in several directions. For instance, Enochs et al. [8, 10] introduce the Gorenstein projective modules and the Gorenstein flat modules; these are analogues of modules of G-dimension 0 for the non-finitely generated arena. Foxby [11], Golod [13] and Vasconcelos [25] focus on finitely generated modules, but consider duality with respect to a semidualizing module. Recently, Holm and Jørgensen [17] have unified these approaches with the G -projective modules and the G -flat modules. For background and definitions, see Sections 2 and 3. The purpose of this paper is to use orsion flat modules in order to further study the G -flat modules, which are more technically challenging to investigate than the G -projective modules. otorsion flat modules have been successfully used to investigate flat modules, for instance in the work of Xu [27], and this paper shows how they are similarly well-suited for studying the G -flat modules. More specifically, an R-module is -flat -orsion when it is isomorphic to an R-module of the form F R where F is flat and orsion. We let F (R) denote the category of all -flat -orsion R-modules, and we let res F (R) denote the category of all R-modules admitting a finite resolution by -flat -orsion R- modules. The first step of our analysis is carried out in Section 4 where we investigate the fundamental properties of these categories; see Theorem I(b) for some of the conclusions from this section. Section 5 contains our analysis of the category of G -flat modules, denoted GF (R). This section culminates in the following theorem. In the terminology of Hashimoto [15], it says that the triple (GF (R), res F (R), F (R)) satisfies the axioms for a weak AB-context. The proof of this result is in (5.9). Theorem I Let be a semidualizing R-module. (a) GF (R) is closed under extensions, kernels of epimorphisms and summands. (b) res F (R) is closed under cokernels of monomorphisms, extensions and summands, and res F (R) res GF (R). (c) F (R) = GF (R) res F (R), and F (R) is an injective cogenerator for GF (R).

3 AB-ontexts and Stability for Gorenstein Flat Modules 405 In conjunction with [15, ( )], this result implies many of the conclusions of [3] for the triple (GF (R), res F (R), F (R)). For instance, we conclude that every module M of finite G -flat dimension fits in an exact sequence 0 Y X M 0 such that X is in GF (R) and Y is in res F (R). Such approximations have been very useful, for instance, in the study of modules of finite G-dimension. See orollary 5.10 for this and other conclusions. InSection 6 we apply these techniques to continue our study of stability properties of Gorenstein categories, initiated in [23]. For each subcategory X of the category of R-modules, let G 1 (X ) denote the category of all R-modules isomorphic to oker( 1 X) for some exact complex X in X such that the complexes Hom R(X, X) and Hom R (X, X ) are exact for each module X in X. This definition is a modification of the construction of G -projective R-modules. Inductively, set G n+1 (X ) = G(G n (X )) for each n 1. The techniques of this paper allow us to prove the following G -flat versions of some results of [23]; see orollary 6.10 and Theorem Theorem II Let be a semidualizing R-module and let n 1. (a) We have G n (GF (R) B (R)) = GF (R) B (R). (b) If dim(r) <, theng n (F (R)) = GF (R) B (R) F (R). Here B (R) is the Bass class associated to, andf (R) is the category of all R-modules N such that Ext 1 R (F R, N) = 0 for each flat R-module F. In particular, when = R this result yields G n (GF(R)) = GF(R) and, when dim(r) is finite, G n (F (R)) = GF(R) F(R). 2 Modules, omplexes and Resolutions We begin with some notation and terminology for use throughout this paper. Definition 2.1 Throughout this work R is a commutative noetherian ring and M(R) is the category of R-modules. We use the term subcategory to mean a full, additive subcategory X M(R) such that, for all R-modules M and N, ifm = N and M X,thenN X. Write P(R), F(R) and I(R) for the subcategories of projective, flat and injective R-modules, respectively. Definition 2.2 We fix subcategories X, Y, W, andv of M(R) such that W X and V Y. Write X Y if Ext 1 R (X, Y) = 0 for each X X and each Y Y. For an R- module M,writeM Y (resp., X M)ifExt 1 R (M, Y) = 0 for each Y Y (resp., if Ext 1 R (X, M) = 0 for each X X ). Set X = the subcategory of R-modules M such that X M. We say W is a cogenerator for X if, for each X X, there is an exact sequence 0 X W X 0

4 406 S. Sather-Wagstaff et al. such that W W and X X ;andw is an injective cogenerator for X if W is a cogenerator for X and X W. Thetermsgenerator and projective generator are defined dually. We say that X is closed under extensions when, for every exact sequence 0 M M M 0 ( ) if M, M X,thenM X. We say that X is closed under kernels of monomorphisms when, for every exact sequence ( ), if M, M X,thenM X. We say that X is closed under cokernels of epimorphisms when, for every exact sequence ( ), if M, M X,thenM X. We say that X is closed under summands when, for every exact sequence ( ), if M X and Eq. splits, then M, M X. We say that X is closed under products when, for every set {M λ } λ of modules in X, we have λ M λ X. Definition 2.3 We employ the notation from [5] for R-complexes. In particular, R- complexes are indexed homologically M = n+1 M M n M n M n 1 M n 1 with nth homology module denoted H n (M). We frequently identify R-modules with R-complexes concentrated in degree 0. Let M, N be R-complexes. For each integer i, let i M denote the complex with ( i M) n = M n i and i M n = ( 1) i n i M.LetHom R(M, N) and M R N denote the associated Hom complex and tensor product complex, respectively. A morphism α : M N is a quasiisomorphism when each induced map H n (α): H n (M) H n (N) is bijective. Quasiisomorphisms are designated by the symbol. The complex M is Hom R (X, )-exact if the complex Hom R (X, M) is exact for each X X. Dually, the complex M is Hom R (, X )-exact if Hom R (M, X) is exact for each X X,andM is R X -exact if M R X is exact for each X X. Definition 2.4 When X n = 0 = H n (X) for all n > 0, the natural morphism X H 0 (X) = M is a quasiisomorphism, that is, the following sequence is exact X + = 2 X 1 X X 1 X 0 M 0. In this event, X is an X -resolution of M if each X n is in X,andX + is the augmented X -resolution of M associated to X. We write projective resolution in lieu of Presolution, and we write flat resolution in lieu of F-resolution. The X -projective dimension of M is the quantity X - pd R (M) = inf{sup{n 0 X n = 0} X is an X -resolution of M}. The modules of X -projective dimension 0 are the nonzero modules of X.Weset res X = the subcategory of R-modules M with X - pd R (M) <. One checks easily that res X is additive and contains X. Following established conventions, we set pd R (M) = P- pd R (M) and fd R (M) = F- pd R (M).

5 AB-ontexts and Stability for Gorenstein Flat Modules 407 The term Y-coresolution is defined dually. The Y-injective dimension of M is denoted Y- id R (M),andtheaugmented Y-coresolution associated to a Y-coresolution Y is denoted + Y. We write injective resolution for I-coresolution, and we set cores Ŷ = the subcategory of R-modules N with Y- id R(N) < which is additive and contains Y. Definition 2.5 A Y-coresolution Y is X -proper if the augmented resolution + Y is Hom R (, X )-exact. We set cores Ỹ = the subcategory of R-modules admitting a Y-proper Y-coresolution. One checks readily that cores Ỹ is additive and contains Y. ThetermY-proper X - resolution is defined dually. Definition 2.6 An X -precover of an R-module M is an R-module homomorphism ϕ : X M where X X such that, for each X X, the homomorphism Hom R (X,ϕ): Hom R (X, X) Hom R (X, M) is surjective. An X -precover ϕ : X M is an X -cover if, every endomorphism f : X X such that ϕ = ϕ f is an automorphism. The terms preenvelope and envelope are defined dually. The next three lemmata have standard proofs; see [3, proofs of (2.1) and (2.3)]. Lemma 2.7 Let 0 M 1 M 2 M 3 0 be an exact sequence of R-modules. (a) If M 3 W,thenM 1 W if and only if M 2 W.IfM 1 W and M 2 W,then M 3 W if and only if the given sequence is Hom R (, W)-exact. (b) If V M 1,thenV M 2 if and only if V M 3.IfV M 2 and V M 3,then V M 1 if and only if the given sequence is Hom R (V, )-exact. (c) If Tor R 1 (M 3, V) = 0, thentor R 1 (M 1, V) = 0 if and only if Tor R 1 (M 2, V) = 0. If Tor R 1 (M 1, V) = 0 = Tor R 1 (M 2, V),thenTor R 1 (M 3, V) = 0 if and only if the given sequence is R V-exact. Lemma 2.8 If X Y, thenx res Ŷ and cores X Y. Lemma 2.9 Let X be an exact R-complex. (a) Assume X i V for all i. If X is Hom R (, V)-exact, then Ker( i X ) V for all i. onversely, if Ker( i X ) V foralliorifx i = 0 for all i 0, thenxis Hom R (, V)-exact. (b) Assume V X i for all i. If X is Hom R (V, )-exact, then V Ker( i X ) for all i. onversely, if V Ker( i X ) foralliorifx i = 0 for all i 0, thenxis Hom R (V, )-exact. (c) Assume Tor1 R (X i, V) = 0 for all i. If the complex X is R V-exact, then Tor1 R (Ker( i X ), V) = 0 for all i. onversely, if Tor1 R (Ker( i X ), V) = 0 foralliorif X i = 0 for all i 0,thenXis R V-exact.

6 408 S. Sather-Wagstaff et al. A careful reading of the proofs of [23, (2.1), (2.2)] yields the next result. Lemma 2.10 Assume that W is an injective cogenerator for X.If M has an X - coresolution that is W-proper and M W, then M is in cores W. 3 ategories of Interest This section contains definitions of and basic facts about the categories to be investigated in this paper. Definition 3.1 An R-module M is orsion if F(R) M.Weset F (R) = the subcategory of flat orsion R-modules. Definition 3.2 The Pontryagin dual or character module of an R-module M is the R-module M = Hom Z (M, Q/Z). One implication in the following lemma is from [27, (3.1.4)], and the others are established similarly. Lemma 3.3 Let M be an R-module. (a) The Pontryagin dual M is R-flat if and only if M is R-injective. (b) The Pontryagin dual M is R-injective if and only if M is R-flat. Semidualizing modules, defined next, form the basis for our categories of interest. Definition 3.4 A finitely generated R-module is semidualzing if the natural homothety morphism R Hom R (, ) is an isomorphism and Ext 1 R (, ) = 0. An R-module D is dualizing if it is semidualizing and has finite injective dimension. Let be a semidualizing R-module. We set P (R) = the subcategory of modules P R where P is R-projective F (R) = the subcategory of modules F R where F is R-flat F (R) = the subcategory of modules F R where F is flat and orsion I (R) = the subcategory of modules Hom R (, I) where I is R-injective. Modules in P (R), F (R), F (R) and I (R) are called -projective, -flat, - flat -orsion, and -injective, respectively. An R-module M is -orsion if F (R) M. Remark 3.5 We justify the terminology -flat -orsion in Lemma 4.3 where we show that M is -flat -orsion if and only if it is -flat and -orsion. The following categories were introduced by Foxby [12], Avramov and Foxby [4], and hristensen [6], though the idea goes at least back to Vasconcelos [25].

7 AB-ontexts and Stability for Gorenstein Flat Modules 409 Definition 3.6 Let be a semidualizing R-module. The Auslander class of is the subcategory A (R) of R-modules M such that (1) Tor1 R (, M) = 0 = Ext1 R (, R M), and (2) The natural map M Hom R (, R M) is an isomorphism. The Bass class of is the subcategory B (R) of R-modules M such that (1) Ext 1 R (, M) = 0 = Tor 1 R (, Hom R(, M)), and (2) The natural evaluation map R Hom R (, M) M is an isomorphism. Fact 3.7 Let be a semidualizing R-module. The categories A (R) and B (R) are closed under extensions, kernels of epimorphisms and cokernels of monomorphism; see [18, or. 6.3]. The category A (R) contains all modules of finite flat dimension and those of finite I -injective dimension, and the category B (R) contains all modules of finite injective dimension and those of finite F -projective dimension by [18, ors. 6.1 and 6.2]. Arguing as in [5, (3.2.9)], we see that M A (R) if and only if M B (R), and M B (R) if and only if M A (R). Similarly, we have M B (R) if and only if Hom R (, M) A (R) by [24, (2.8.a)]. From [18, Thm. 6.1] we know that every module in B (R) has a P -proper P -resolution. The next definitions are due to Holm and Jørgensen [17] in this generality. Definition 3.8 Let be a semidualizing R-module. A complete I I-resolution is a complex Y of R-modules satisfying the following: (1) Y is exact and Hom R (I, )-exact, and (2) Y i is -injective when i 0 and Y i is injective when i < 0. An R-module H is G -injective if there exists a complete I I-resolution Y such that H = oker( Y 1 ),inwhichcasey is a complete I I-resolution of H.Weset GI (R) = the subcategory of G -injective R-modules. In the special case = R, wewritegi(r) in place of GI R (R). A complete FF -resolution is a complex Z of R-modules satisfying the following. (1) Z is exact and R I -exact. (2) Z i is flat if i 0 and Z i is -flat if i < 0. An R-module M is G -flat if there exists a complete FF -resolution Z such that M = oker( Z 1 ),inwhichcasez is a complete FF -resolution of M. Weset GF (R) = the subcategory of G -flat R-modules. In the special case = R, wesetgf(r) = GF R (R), andgfd = GF- pd. A complete PP -resolution is a complex X of R-modules satisfying the following. (1) X is exact and Hom R (, P )-exact. (2) X i is projective if i 0 and X i is -projective if i < 0.

8 410 S. Sather-Wagstaff et al. An R-module M is G -projective if there exists a complete PP -resolution X such that M = oker( X 1 ), in which case X is a complete PP -resolution of M.Weset GP (R) = the subcategory of G -projective R-modules. Fact 3.9 Let be a semidualizing R-module. Flat R-modules and -flat R-modules are G -flat by [17, (2.8.c)]. It is straightforward to show that an R-module M is G - flat if and only the following conditions hold: (1) M admits an augmented F -coresolution that is R I -exact, and (2) Tor R 1 (M, I ) = 0. Let R denote the trivial extension of R by, defined to be the R-module R R = R with ring structure given by (r, c)(r, c ) = (rr, rc + r c). EachRmodule M is naturally an R -module via the natural surjection R R. Within this protocol we have M GI (R) if and only if M GI(R ) and M GF (R) if and only if M GF(R ) by [17, (2.13) and (2.15)]. Also [17, (2.16)] implies GF - pd R (M) = Gfd R (M). The next definition, from [23], is modeled on the construction of GI(R). Definition 3.10 Let X be a subcategory of M(R). A complete X -resolution is an exact complex X in X that is Hom R (X, )-exact and Hom R (, X )-exact. 1 Such a complex is a complete X -resolution of oker( X 1 ).Weset G(X ) = the subcategory of R-modules with a complete X -resolution. Set G 0 (X ) = X, G 1 (X ) = G(X ) and G n+1 (X ) = G(G n (X )) for n 1. Fact 3.11 Let X be a subcategory of M(R). Using a resolution of the form 0 X 0, one sees that X G(X ) and so G n (X ) G n+1 (X ) for each n 0. If is a semidualizing R-module, then G n (I (R)) = GI (R) A (R) for each n 1; see [23, (4.4)]. The final definition of this section is for use in the proof of Theorem II. Definition 3.12 Let be a semidualizing R-module, and let X be a subcategory of M(R). AP F -complete X -resolution is an exact complex X in X that is Hom R (P, )-exact and Hom R (, F )-exact. Such a complex is a P F -complete X -resolution of oker( 1 X).Weset H (X ) = the subcategory of R-modules with a P F -complete X -resolution. Set H 0 (X ) = X, H1 (X ) = H (X ) and H n+1 (X ) = H (H n (X )) for each n 1. 1 In the literature, these complexes are sometimes called totally acyclic.

9 AB-ontexts and Stability for Gorenstein Flat Modules 411 Remark 3.13 Let be a semidualizing R-module, and let X be a subcategory of M(R). Let X be an exact complex in X that is Hom R (, )-exact and Hom R (, F )-exact. Hom-tensor adjointness implies that X is Hom R(P, )-exact and hence a P F -complete X -resolution, as is the complex i X for each i Z. It follows that oker( i X ) H (X ) for each i. Using a resolution of the form 0 X 0, one sees that X H (X ) and so H n (X ) Hn+1 (X ) for each n 0. Furthermore, if F (R) X,thenG(X) H (X ) and so G n (X ) H n (X ) for each n 1. 4 Modules of Finite F -projective Dimension This section contains the fundamental properties of the modules of finite F - projective dimension. The first two results allow us to deduce information for these modules from the modules of finite I (R)-injective dimension. Lemma 4.1 Let M be an R-module, and let be a semidualizing R-module. (a) The Pontryagin dual M is -flat if and only if M is -injective. (b) The Pontryagin dual M is -injective if and only if M is -flat. (c) If Tor R 1 (, M) = 0, thenm is -orsion. (d) If M is -injective, then M is -flat and -orsion. Proof (a) Assume that M is -injective, so there exists an injective R-module I such that M = Hom R (, I). This yields the first isomorphism in the following sequence while the second is from Hom-evaluation [7, Prop. 2.1(ii)]: M = HomZ (Hom R (, I), Q/Z) = R Hom Z (I, Q/Z). Since I is injective, Lemma 3.3(b) implies that Hom Z (I, Q/Z) is flat. Hence, the displayed isomorphisms imply that M is -flat. onversely, assume that M is -flat, so there exists a flat R-module F such that M = F R.AsF isflatitisina (R), and this yields the first isomorphism in the next sequence, while the third isomorphism is Hom-tensor adjointness F = Hom R (, F R ) = Hom R (, Hom Z (M, Q/Z)) = Hom Z ( R M, Q/Z). This module is flat, and so Lemma 3.3(a) implies that R M is injective. From [18, Thm. 1] we conclude that M is -injective. (b) This is proved similarly. (c) Let P be a projective resolution of M. Our Tor-vanishing hypothesis implies that there is a quasiisomorphism R P R M. For each flat R-module F, this yields a quasiisomorphism F R R P F R R M.

10 412 S. Sather-Wagstaff et al. BecauseQ/Z is injective over Z, this provides the third quasiisomorphism in the next sequence, while the second quasiisomorphism is Hom-tensor adjointness Hom R (F R, P ) Hom R (F R, Hom Z (P, Q/Z)) Hom Z (F R R P, Q/Z) Hom Z (F R R M, Q/Z). Since Q/Z is injective over Z, there are quasiisomorphisms M Hom Z (M, Q/Z) Hom Z (P, Q/Z) P. By Lemma 3.3(a), it follows that P is an injective resolution of M over R. In particular, taking cohomology in the displayed sequence ( ) yields isomorphisms Ext i R (F R, M ) = H i (Hom R (F R, P )) = H i (Hom Z (F R R M, Q/Z)). This is 0 when i = 0 because Hom Z (F R R M, Q/Z) is a module. Hence, the desired conclusion. (d) Since M is -injective, it is in A (R) by Fact 3.7, and so Tor1 R (, M) = 0. Hence M is -orsion by part (c), and it is -flat by part (a). Lemma 4.2 Let M be an R-module, and let be a semidualizing R-module. (a) There is an equality I - id R (M ) = F - pd R (M). (b) There is an equality F - pd R (M ) = I - id R (M). Proof We prove part (a); the proof of part (b) is similar. For the inequality I - id R (M ) F - pd R (M), assume that F - pd R (M) <.Let X be a F (R)-resolution of M such that X i = 0 for all i > F - pd R (M). It follows from Lemma 4.1(b) that the complex X is an I -coresolution of M such that X i = 0 for all i > F - pd R (M). The desired inequality now follows. For the reverse inequality, assume that j = I - id R (M )<. Fact 3.7 implies that M is in A (R), and hence also implies that M B (R). This condition implies that M has a proper P -resolution Z by Fact 3.7. In particular, this is an F -resolution of M, so Lemma 4.1(b) implies that Z is an I -coresolution of M. We claim that Z is a proper I -coresolution of M.LetI be an injective R- module. By assumption, the complex Hom R (, Z + ) is exact. Since Q/Z is an injective Z-module, we have (Z ) + = (Z + ) = Hom Z (Z +, Q/Z), and this explains the first isomorphism in the next sequence Hom R ((Z ) +, Hom R (, I)) = Hom R (Hom Z (Z +, Q/Z), Hom R (, I)) = Hom R ( R Hom Z (Z +, Q/Z), I) = Hom R (Hom Z (Hom R (, Z + ), Q/Z), I). The second isomorphism is Hom-tensor adjointness, and the third isomorphism is Hom-evaluation [7, Prop. 2.1(ii)]. Since Hom R (, Z + ) is exact, we conclude that the complex Hom R (Hom Z (Hom R (, Z + ), Q/Z), I) is also exact because Q/Z is an injective Z-module and I is an injective R-module. This shows that (Z ) + is Hom R (, I )-exact, and establishes the claim. ( )

11 AB-ontexts and Stability for Gorenstein Flat Modules 413 From [24, (3.3.b)] we know that Ker(( j+1 Z ) ) = oker( j+1 Z ) is in I (R). Lemma 4.1(b) implies oker( j+1 Z ) F (R). It follows that the truncated complex Z : 0 oker( j+1 Z ) Z j 1 Z 0 0 is an F -resolution of M such that Z i = 0 for all i > j. The desired inequality now follows, and hence the equality. The next three lemmata document properties of F (R) for use in the sequel. The first of these contains the characterization of -flat -orsion modules mentioned in Remark 3.5. Lemma 4.3 Let and M be R-modules with semidualizing. The following conditions are equivalent: (i) M F (R); (ii) M F (R) and F (R) M; (iii) M B (R) and Hom R (, M) F (R); (iv) Hom R (, M) F (R). In particular, we have F (R) F (R). Proof (i) (ii). It suffices to show, for each flat R-module F, thatf(r) F if and only if F (R) F R.LetF be a flat R-module. It suffices to show that Ext i R (F R, F R ) = Ext i R (F, F) for each i. From[26, (1.11.a)] we have the first isomorphism in the next sequence Ext i R (, F R ) = Ext i R (, ) R F = { R R F = F if i = 0 0 R F = 0 if i = 0 and the second isomorphism is from the fact that is semidualizing. Let P be a projective resolution of. The previous display provides a quasiisomorphism Hom R (P, F R ) F. Let P be a projective resolution of F. Hom-tensor adjointness yields the first quasiisomorphism in the next sequence Hom R (P R P, F R ) Hom R (P, Hom R (P, F R )) Hom R (P, F) and the second quasiisomorphism is from the previous display, because P is a bounded below complex of projective R-modules. Since F is flat, we conclude that

12 414 S. Sather-Wagstaff et al. P R P is a projective resolution of F R. It follows that we have Ext i R (F R, F R ) = H i (Hom R (P R P, F R )) = H i (Hom R (P, F)) = Ext i R (F, F) as desired. (i) = (iii). Assume that M F (R), thatis,thatm = R F for some F F (R) A (R).Then Hom R (, M) = Hom R (, R F) = F F (R) and M F (R) F (R) B (R). (iii) = (i). If M B (R) and Hom R (, M) F (R), then there is an isomorphism M = R Hom R (, M) F (R). (iii) (iv). This is from Fact 3.7 because F (R) A (R). The conclusion F (R) F (R) follows from the implication (i) = (ii). Lemma 4.4 If is a semidualzing R-module, then the category F (R) is closed under products, extensions and summands. Proof onsider a set {F λ } λ of modules in F (R). From[9, (3.2.24)] we have λ F λ F (R) and so R ( λ F λ) F (R). Hence, we have λ ( R F λ ) = R ( λ F λ) F (R) where the isomorphism comes from the fact that is finitely presented. Thus F (R) is closed under products. By Lemma 2.7(b), the category of -orsion R-modules is closed under extensions, and it is closed under summands by the additivity of Ext. The category F (R) is closed under extensions and summands by [18, Props. 5.1(a) and 5.2(a)]. The result now follows from Lemma 4.3. Note that the hypotheses of the next lemma are satisfied when M F (R) B (R). Lemma 4.5 Let be a semidualizing R-module, and let M be a -orsion R- module such that the natural evaluation map R Hom R (, M) M is bijective. (a) The module M has an F -cover, and every -flat cover of M is an F -cover of M with -orsion kernel. (b) Each F -precover of M is surjective. (c) Assume further that Tor1 R (, Hom R(, M)) = 0. Then M has an F -proper F - resolution such that Ker( i 1 X ) is -orsion for each i. Proof (a) The module M has a -flat cover ϕ : F R M by [18, Prop. 5.3(a)], and Ker(ϕ) is -orsion by [27, (2.1.1)]. Furthermore, the bijectivity of the evaluation map R Hom R (, M) M implies that there is a projective R-module P and a surjective map ϕ : P R M by [24, (2.2.a)]. The fact that ϕ is a precover

13 AB-ontexts and Stability for Gorenstein Flat Modules 415 provides a map f : P R F R such that ϕ = ϕ f. Hence, the surjectivity of ϕ implies that ϕ is surjective. It follows from Lemma 2.7(a) that F R is - orsion, and so F R F (R) by Lemma 4.3. Since ϕ is a -flat cover and F (R) F (R), we conclude that ϕ is an F -cover. (b) This follows as in part (a) because M has a surjective F -cover. (c) Using parts (a) and (b), the argument of [18, Thm. 2] shows how to construct a resolution with the desired properties. The final three results of this section contain our main conclusions for res F (R). The first of these extends Lemma 4.3. Proposition 4.6 Let and M be R-modules with semidualizing, and let n 0.The following conditions are equivalent: (i) F - pd R (M) n; (ii) M B (R) and F - pd R (Hom R (, M)) n; (iii) F - pd R (Hom R (, M)) n; (iv) M = R K for some R-module K such that F - pd R (K) n; (v) F - pd R (M) nandf (R) M. Proof (i) = (ii) Since F - pd R (M) n <, we have M B (R) by Fact 3.7. Let X be an F -resolution of M such that X i = 0 when i > n. for each i,letf i F (R) such that X i = Fi R. Since each F i is in A (R), we have Hom R (, X) i = HomR (, X i ) = Hom R (, F i R ) = F i. A standard argument using the conditions M, X i B (R) shows that Hom R (, X) is an F -resolution of Hom R (, M) such that Hom R (, X) i = 0 when i > n. The inequality F - pd R (Hom R (, M)) n then follows. (ii) = (iv) The condition M B (R) implies M = R Hom R (, M), andso K = Hom R (, M) satisfies the desired conclusions. (iv) = (v) Let F be an F -resolution of K such that F i = 0 when i > n. Using the condition K, F i A (R), a standard argument shows that R F is an F -resolution of R K = M. Hence, this resolution yields F - pd R (M) F - pd R (M) n. By Lemma 4.3, we have F (R) F (R), and so Lemma 2.8 implies F (R) res F (R); inparticularf (R) M. (v) = (i) The assumption F - pd R (M) n implies M B (R) by Fact 3.7, and so Ext 1 R (, M) = 0. Lemma 4.5(c) implies that M has an F -proper F -resolution X such that K i = Ker( i 1 X ) is -orsion for each i. In particular, the truncated complex X = 0 K n X n 1 X 0 M 0 is exact and Hom R (, )-exact. Since F - pd R (M) n, the proof of the implication (i) = (ii) shows that fd R (Hom R (, M)) n. Since each R-module Hom R (, X i ) is flat by Lemma 4.3, the exact complex Hom R (, X ) is a truncation of an augmented flat resolution of Hom R (, M). It follows that Hom R (, K n ) is flat, and so K n F (R) by [18, Thm. 1]. Hence X is an augmented F -resolution of M, andso F - pd R (M) n. (ii) (iii) follows from Fact 3.7 because res F (R) A (R).

14 416 S. Sather-Wagstaff et al. Lemma 4.7 Let be a semidualizing R-module. If F - pd R (M) <, thenany bounded F -resolution X of M is F -proper. Proof Observe that F (R) X i for all i and F (R) M by Proposition 4.6. So, the complex X + is exact and such that (X + ) i = 0 for i 0 and F (R) (X + ) i.hence, Lemma 2.9(b) implies that X + is Hom R (F, )-exact. Proposition 4.8 Let be a semidualizing R-module. The category res F (R) is closed under extensions, cokernels of monomorphisms and summands. Proof onsider an exact sequence 0 M 1 M 2 M 3 0 such that F - pd R (M 1) and F - pd R (M 3) are finite. To show that res F (R) is closed under extensions we need to show that F - pd R (M 2) is finite. The condition F - pd R (M 1)< implies I - id(m1 ) = F - pd R (M 1 )< by Lemma 4.2(a) and Proposition 4.6; and similarly I - id(m3 )<. From[24, (3.4)] we know that the category of R-modules of finite I -injective dimension is closed under extensions. Using the dual exact sequence 0 M 3 M 2 M 1 0 we conclude that I - id(m2 ) is finite. Lemma 4.2(a) implies that F - pd R (M 2 ) is finite. Since F - pd R (M 1)<, Proposition 4.6 implies F (R) M 1 ; and similarly F (R) M 3. Thus, we have F (R) M 2 by Lemma 2.7(b). ombining this with the previous paragraph, Proposition 4.6 implies that F - pd R (M 2)<. The proof of the fact that res F (R) is closed under cokernels of monomorphisms is similar. The fact that res F (R) is closed under summands is even easier to prove using the natural isomorphism (M 1 M 2 ) = M 1 M2. 5 Weak AB-ontext Let be a semidualizing R-module. The point of this section is to show that the triple (GF (R), res F (R), F (R)) is a weak AB-context, and to document the immediate consequences; see Theorem I and orollary We begin the section with two results modeled on [16, (3.22) and (3.6)]. F Lemma 5.1 If is a semidualizing R-module, then GF (R) res (R). Proof By Lemma 2.8 it suffices to show GF (R) F (R). Fix modules M GF (R) and N F (R). By Lemma 4.1, we know that the Pontryagin dual N is -injective. Hence, for i 1, the vanishing in the next sequence is from Fact 3.9 Ext i R (M, N ) = Ext i R (M, Hom Z(N, Q/Z)) = Hom Z (Tor i R (M, N ), Q/Z) = 0. The second isomorphism is a form of Hom-tensor adjointness using the fact that Q/Z is injective over Z. To finish the proof, it suffices to show that N is a summand

15 AB-ontexts and Stability for Gorenstein Flat Modules 417 of N ; then the last sequence shows Ext 1 R (M, N) = 0. Write N = R F for some flat orsion R-module F, and use Hom-tensor adjointness to conclude N = HomZ ( R F, Q/Z) = Hom R (, Hom Z (F, Q/Z)). Lemma 3.3(b) implies that Hom Z (F, Q/Z) is injective, so the proof of Lemma 4.1(a) explains the second isomorphism in the next sequence N = HomR (, Hom Z (F, Q/Z)) = R Hom Z (Hom Z (F, Q/Z), Q/Z) = R F. The proof of [16, (3.22)] shows that F is a summand of F, and it follows that N = R F is a summand of R F = N, as desired. Lemma 5.2 Let be a semidualizing R-module. If M is an R-module, then M is in GF (R) if and only if its Pontryagin dual M is in GI (R). Proof onsider the trivial extension R from Fact 3.9. By [16, (3.6)] we know that M is in GF(R ) if and only if M is in GI(R ). AlsoM is in GF(R ) if and only if M is in GF (R),andM is in GI(R ) if and only if M is in GI (R) by Fact 3.9. Hence, the equivalence. The following result establishes Theorem I(a). Proposition 5.3 Let be a semidualizing R-module. The category GF (R) is closed under kernels of epimorphisms, extensions and summands. Proof The result dual to [26, (2.8)] says that GI (R) is closed under cokernels of monomorphisms, extensions and summands. To see that GF (R) is closed under summands, let M GF (R) and assume that N is a direct summand of M. It follows that the Pontryagin dual N is a direct summand of M. Lemma 5.2 implies that M is in GI (R) which is closed under summands. We conclude that N GI (R), and so N GF (R).HenceGF (R) is closed under summands, and the other properties are verified similarly. The next four results put the finishing touches on Theorem I. Lemma 5.4 Let be a semidualizing R-module. If X is a complete FF -resolution, then oker( X n ) GF (R) for each n Z. Proof Write M n = oker( X n ), and note that M 1 GF (R) by definition. Fact 3.9 implies that X n GF (R) for each n Z. SinceM 1 is in GF (R), an induction argument using Proposition 5.3 shows M n GF (R) for each n 1. Now assume n 0. Lemma 2.9(c), implies Tor R 1 (M n, I ) = 0. By construction, the following sequence is exact and R I -exact 0 M n X n 2 X n 3 with each X n i GF (R), andsom n GF (R) by Fact 3.9. Lemma 5.5 Let be a semidualizing R-module. If M F (R), then there is an exact sequence 0 M M 1 M 2 0 with M 1 F (R) and M 2 F (R).

16 418 S. Sather-Wagstaff et al. Proof Since M is -flat, we know from [18, Thm. 1] that Hom R (, M) is flat. By [27, (3.1.6)] there is a orsion flat module F containing Hom R (, M) such that the quotient F/ Hom R (, M) is flat. onsider the exact sequence 0 Hom R (, M) F F/ Hom R (, M) 0. Since F/ Hom R (, M) is flat, an application of R yields an exact sequence 0 R Hom R (, M) R F R (F/ Hom R (, M)) 0. Because M is -flat, it is in B (R) and so R Hom R (, M) = M. WithM 1 = R F and M 2 = R (F/ Hom R (, M)) this yields the desired sequence. Lemma 5.6 Let be a semidualizing R-module. Each module M GF (R) admits an injective F -preenvelope α : M Y such that oker(α) GF (R). Proof Let M GF (R) with complete FF -resolution X. By definition, this says that M is a submodule of the -flat R-module X 1, and Lemma 5.4 implies that X 1 /M GF (R).SinceX 1 is -flat, Lemma 5.5 yields an exact sequence 0 X 1 Z Z / X 1 0 with Z F (R) and Z / X 1 F (R). It follows that Z / X 1 is in GF (R). Since X 1 /M is also in GF (R),andGF (R) is closed under extensions by Proposition 5.3, the following exact sequence shows that Z /M is also in GF (R) 0 X 1 /M Z /M Z / X 1 0. In particular, Lemma 5.1 implies Z /M F (R), and it follows that the next sequence is Hom R (, F )-exact by Lemma 2.7(a). 0 M R F Z /M 0 The conditions Z F (R) and Z /M GF (R) then implies that the inclusion M Z is an F -preenvelope whose cokernel is in GF (R). Proposition 5.7 Let be a semidualizing R-module. The category F (R) is an injective cogenerator for the category GF (R). In particular, every module in GF (R) admits a F -proper F -coresolution, and so GF (R) cores (R). F Proof Lemmas 5.1 and 5.6 imply that F (R) is an injective cogenerator for GF (R). The remaining conclusions follow immediately. Lemma 5.8 If is a semidualizing R-module, then there is an equality F (R) = GF (R) res (R). F Proof The containment F (R) GF (R) res F (R) is straightforward; see Definition 2.4 and Fact 3.9. For the reverse containment, let M GF (R) res (R). Truncate a bounded F -resolution to obtain an exact sequence F 0 K F R M 0

17 AB-ontexts and Stability for Gorenstein Flat Modules 419 with F F (R) and such that F - pd R (K) <. We have Ext1 R (M, K) = 0 by Lemma 5.1, so this sequence splits. Hence M is a summand of F R F (R). Lemma 4.4 implies that F (R) is closed under summands, so M F (R). 5.9 Proof of Theorem 1 Part (a) is in Proposition 5.3. SinceF (R) GF (R) by Fact 3.9, we have res F (R) res GF (R). With this, part (b) follows from Proposition 4.8. Proposition 5.7 and Lemma 5.8 justify part (c). Here is the list of immediate consequences of Theorem I and [15, ( )]. For part (a), recall that add(x ) is the subcategory of all R-modules isomorphic to a direct summand of a finite direct sum of modules in X. orollary 5.10 Let be a semidualizing R-module and let M res GF (R). (a) If X is an injective cogenerator for GF (R),thenadd(X ) = F (R). (b) There exists an exact sequence 0 Y X M 0 with X GF (R) and Y res F (R). (c) There exists an exact sequence 0 M Y X 0 with X GF (R) and Y res F (R). (d) The following conditions are equivalent: (i) M GF (R); (ii) Ext 1 R (M, res F ) = 0; (iii) Ext 1 R (M, res F ) = 0; (iv) Ext 1 R (M, F ) = 0. Thus, the surjection X Mfrom(b)isaGF -precover of M. (e) The following conditions are equivalent: (i) M res F (R); (ii) Ext 1 R (GF, M) = 0; (iii) Ext 1 R (GF, M) = 0; (iv) sup{i 0 Ext i R (GF, M) = 0} < and Ext 1 R (F, M) = 0. Thus, the injection M Yfrom(c)isares F -preenvelope of M. (f) There are equalities GF - pd R (M) = sup{i 0 Ext i R (M, res F ) = 0} = sup{i 0 Ext i R (M, F ) = 0} (g) There is an inequality GF - pd R (M) F - pd R (M) with equality when F - pd R (M) <. (h) The category res GF (R) is closed under extensions, kernels of epimorphisms and cokernels of monomorphisms. For the next result recall that the triple (GF (R), res F (R), F (R)) is an ABcontext if it is a weak AB-context and such that res GF (R) = M(R).

18 420 S. Sather-Wagstaff et al. Proposition 5.11 Assume that dim(r) is finite, and let be a semidualizing R-module. The triple (GF (R), res F (R), F (R)) is an AB-context if and only if is dualizing for R. Proof Assume first that (GF (R), res F (R), F (R)) is an AB-context. Recall that every maximal ideal of the trivial extension R is of the form m for some maximal ideal m R, and there is an isomorphism (R )/(m ) = R/m. With Fact 3.9, this yields the equality in the next sequence Gfd (R )m ((R ) m /(m ) m ) Gfd R ((R )/(m )) = GF - pd R (R/m) <. The first inequality follows from [5, (5.1.3)], and the finiteness is by assumption. Using [5, (1.2.7),(1.4.9),(5.1.11)] we deduce that the following ring is Gorenstein (R ) m = Rm m and so [21, (7)] implies that m is dualizing for R m. (This also follows from [6, (8.1)] and [17, (3.1)].) Since this is true for each maximal ideal of R and dim(r) <, we conclude that is dualizing for R by [14, (5.8.2)]. onversely, assume that is dualizing for R. Using Theorem I, it suffices to show that each R-module M has GF - pd R (M) <. Since is dualizing, the trivial extension R is Gorenstein by [21, (7)]. Also, we have dim(r ) = dim(r) < as Spec(R ) is in bijection with Spec(R). Thus, in the next sequence GF - pd R (M) = Gfd R (M) < the finiteness is from [9, (12.3.1)] and the equality is from Fact 3.9. To end this section, we prove a complement to [26, (3.6)] which establishes the existence of certain approximations. For this, we need the following preliminary result which compares to Lemma 5.8. Lemma 5.12 If is a semidualizing R-module, then there is an equality F (R) = GF (R) res F (R). Proof The containment F (R) GF (R) res F (R) is from Definition 2.4 and Fact 3.9. For the reverse containment, let M GF (R) res F (R). Letn 1 be an integer with F - pd R (M) n. We show by induction on n that M is -flat. For the base case n = 1, there is an exact sequence 0 X 1 X 0 M 0 ( ) with X 1, X 0 F (R). Lemma 5.5 provides an exact sequence 0 X 1 Y 1 Y 2 0 ( )

19 AB-ontexts and Stability for Gorenstein Flat Modules 421 with Y 1 F (R) and Y 2 F (R). onsider the following pushout diagram whose top row is Eq. and whose leftmost column is Eq X 1 X 0 M 0 0 Y 1 V = M 0 ( ) Y 2 = Y Since M is in GF (R) and Y 1 is in F (R), Lemma 5.1 implies Ext1 R (M, Y 1) = 0. Hence, the middle row of Eq. splits. The subcategory F (R) is closed under extensions and summands by [18, Props. 5.1(a) and5.2(a)]. Hence, the middle column of Eq. shows that V F (R), so the fact that the middle row of Eq. splits implies that M F (R), as desired. For the induction step, assume that n 2. Truncate a bounded F -resolution of M to find an exact sequence 0 K Z M 0 such that Z F (R) and F - pd R (K) n 1. By induction, we conclude that K F (R). Hence, the displayed sequence implies F - pd R (M) 1, and the base case implies that M F (R). Proposition 5.13 Let be a semidualizing R-module and assume that dim(r) is finite. If M GF (R), then there exists an exact sequence such that K F (R) and X GP (R). 0 K X M 0 Proof Since M is in GF (R) and dim(r) <, we know that GP - pd R (M) < by [22, (3.3.c)]. Hence, from [26, (3.6)] there is an exact sequence 0 K X M 0 with K res P (R) and X GP (R). From[22, (3.3.a)] we have X GP (R) GF (R). SinceGF (R) is closed under kernels of epimorphisms by Proposition 5.3, the displayed sequence implies that K GF (R). The containment P (R) F (R) implies K res P (R) res F (R), and so Lemma 5.12 says K F (R). Thus, the displayed sequence has the desired properties.

20 422 S. Sather-Wagstaff et al. 6 Stability of ategories This section contains our analysis of the categories G n (F (R)) and G n (F (R)); see Definition We draw many of our conclusions from the known behavior for G n (I (R)) using Pontryagin duals. This requires, however, the use of the categories H n (F (R)) and H n (R)) as a bridge; see Definition (F Lemma 6.1 Let be a semidualizing R-module, and let X be an R-complex. If X is Hom R (, F )-exact, then it is R I -exact. Proof Let N I (R). From Lemmas 4.1(d) and 4.3 we know that the Pontryagin dual N is in F (R). Hence, the following complex is exact by assumption Hom R (X, N ) = Hom R (X, Hom Z (N, Q/Z)) = Hom Z (X R N, Q/Z). As Q/Z is faithfully injective over Z, we conclude that X R N is exact, and so X is R I -exact. Note that the hypotheses of the next lemma are satisfied whenever X GF (R) by Fact 3.9 and Lemma 5.1. Lemma 6.2 Let be a semidualizing R-module and X a subcategory of M(R). (a) If Tor1 R (X, I ) = 0, thentor1 R (Hn (X ), I ) = 0 for each n 1. (b) If X F (R), thenhn (X ) F (R) for each n 1. Proof By induction on n, it suffices to prove the result for n = 1. We prove part (a). The proof of part (b) is similar. Let M H (X ) with P F -complete X -resolution X. ThecomplexX is R I -exact by Lemma 6.1. Since we have assumed that Tor1 R (X, I ) = 0, the desired conclusion follows from Lemma 2.9(c) because M = Ker( 1 X ). The converse of the next result is in Proposition 6.5. Lemma 6.3 If is a semidualizing R-module and M H (F (R)), then M G(I (R)). Proof Let X be a P F -complete F -resolution of M. Lemma 4.1(b) implies that the complex X = Hom Z (X, Q/Z) is an exact complex in I (R). Furthermore M = oker( 1 X ). Thus, it suffices to show that X is Hom R (I, )-exact and Hom R (, I )- exact. Let I be an injective R-module. The second isomorphism in the next sequence is Hom-evaluation [7, Prop. 2.1(ii)] R X = R Hom Z (X, Q/Z) = Hom Z (Hom R (, X), Q/Z). Since Hom R (, X) is exact by assumption, we conclude that R X = X R is also exact. It follows that the following complexes are also exact Hom R (X R, I) = Hom R (X, Hom R (, I)) where the isomorphism is Hom-tensor adjointness. Thus X is Hom R (, I )-exact.

21 AB-ontexts and Stability for Gorenstein Flat Modules 423 Lemma 6.1 implies that the complex Hom R (, I) R X is exact. Hence, the following complexes are also exact Hom Z (Hom R (, I) R X, Q/Z) = Hom R (Hom R (, I), Hom Z (X, Q/Z)) = Hom R (Hom R (, I), X ) and so X is Hom R (I, )-exact. The next result is a version of [23, (5.2)] for H (F (R)). Proposition 6.4 If is a semidualizing R-module, then there is an equality H (F (R)) = GF (R) B (R). Proof For the containment H (F (R)) GF (R) B (R), letm H (F (R)), and let X be a P F -complete F -resolution of M. Lemma 6.1 implies that X is R I -exact, and so the sequence 0 M X 1 X 2 satisfies condition 3.9(1). Fact 3.9 implies Tor1 R (F, I ) = 0 and so Lemma 6.2(a) provides Tor1 R (M, I ) = 0. From Fact 3.9 we conclude M GF (R). Also, Lemma 6.3 guarantees that M G(I (R)), andsom A (R) by Fact Thus, Fact 3.7 implies M B (R). For the reverse containment, let M GF (R) B (R), andlety be a complete FF -resolution of M. In particular, the complex 0 M Y 1 Y 2 ( ) is an augmented F -coresolution of M and is R I -exact. We claim that this complex is also Hom R (, )-exact and Hom R (, F )-exact. For each i Z set M i = oker( i Y ). This yields an isomorphism M = M 1. By assumption, we have M, Y i B (R) for each i < 0, andso M and Y i. Thus, Lemma 2.8(b) implies that the complex ( ) ishom R (, )-exact. From Lemma 5.4 we conclude M i GF (R) for each i, andsom i F (R) by Lemma 5.1. Lemma 4.3 implies Y i F (R) for each i < 0, and so Lemma 2.9(a) guarantees that Eq. is also Hom R (, F )-exact. Because M B (R), Fact 3.7 provides an augmented P -proper P -resolution 2 Z 1 Z Z 1 Z 0 M 0. ( ) Since each Z i P (R) F (R), we have Z i F (R) by Lemma 4.3. Since M F (R), we see from Lemma 2.9(a) that Eq. is also Hom R(, F )-exact. It follows that the complex obtained by splicing the sequences ( ) and( ) isa P F -complete F -resolution of M. Thus M H (F (R)), as desired. Our next result contains the converse to Lemma 6.3. Proposition 6.5 Let be a semidualizing R-module and M an R-module. Then M H (F (R)) if and only if M G(I (R)).

22 424 S. Sather-Wagstaff et al. Proof One implication is in Lemma 6.3. For the converse, assume that M is in G(I (R)) = GI (R) A (R); see Fact Fact 3.7 and Lemma 5.2 combine with Proposition 6.4 to yield M B (R) GF (R) = H (F (R)). The next three lemmata are for use in Theorem 6.9. Lemma 6.6 If is a semidualizing R-module, then H 2 (F (R)) B (R). Proof Let M H 2 (F (R)) and let X be a P F -complete H (F )-resolution of M. In particular, the complex Hom R (, X) is exact. Each module X i is in H (F (R)) B (R) by Proposition 6.4, and so Ext 1 R (, X i) = 0 for each i. Thus, Lemma 2.8(b) implies that Ext 1 R (, M) = 0. Also, since M = Ker( 1 X ), the leftexactness of Hom R (, ) implies that Hom R (, M) = Ker( HomR(,X) 1 ). The natural evaluation map R Hom R (, X i ) X i is an isomorphism for each i because X i B (R), and so we have R Hom R (, X) = X. In particular, the complex Hom R (, X) is R -exact. As Tor1 R (, Hom R(, X i )) = 0 for each i, Lemma 2.9(c) implies that Tor1 R (, Hom R(, M)) = 0. Finally, each row in the following diagram is exact R Hom R (, X 1 ) R Hom R (, X 0 ) R Hom R (, M) 0 = = X 1 X 0 M 0 and the vertical arrows are the natural evaluation maps. A diagram chase shows that the rightmost vertical arrow is an isomorphism, and so M B (R). Lemma 6.7 If is a semidualizing R-module, then F (R) is an injective cogenerator for H (F (R)). Proof The containment in the following sequence is from Facts 3.7 and 3.9 F (R) GF (R) B (R) = H (F (R)) and the equality is from Proposition 6.4. Lemma 5.1 implies GF (R) F (R). Thus, the conditions H (F (R)) = GF (R) B (R) GF (R) imply that we have H (F (R)) F (R). Let M H (F (R)) GF (R). SinceF (R) is an injective cogenerator for GF (R) by Proposition 5.7, there is an exact sequence 0 M X M 0 with X F (R) and M GF (R). SinceM and X are in B (R), Fact 3.7 implies that M B (R). ThatisM GF (R) B (R) = H (F (R)). This establishes the desired conclusion. F Lemma 6.8 If is a semidualizing R-module, then H 2 (F (R)) cores (R).

23 AB-ontexts and Stability for Gorenstein Flat Modules 425 Proof Lemma 6.7 says that F (R) is an injective cogenerator for H (F (R)). By Lemma 6.2(b) we know that H 2 (F (R)) F (R). LetM H2 (F (R)) and let X be a P F -complete H (F )-resolution of M. By definition, the complex 0 M X 1 X 2 is an augmented H (F )-coresolution that is F -proper and therefore F -proper. Hence, Lemma 2.10 implies M cores (R). F Theorem II For each semidualizing R-module and each integer n 1, thereisan equality H n (F (R)) = GF (R) B (R). Proof We first verify the equality H 2 (F (R)) = H (F (R)). Remark 3.13 implies H 2 (F (R)) H (F (R)). For the reverse containment, let M H 2 (F (R)). Lemma 4.3 implies F (R) F (R), andsom F (R) by Lemma 6.2(b). From Lemma 6.6 we have M B (R), and so Fact 3.7 provides an augmented P -proper P -resolution 2 Z 1 Z Z 1 Z 0 M 0. ( ) Each Z i P (R) F (R), so we have Z i F (R) by Lemma 4.3. We conclude from Lemma 2.9(a) that Eq. is Hom R (, F )-exact. Lemma 6.8 yields a F -proper augmented F -coresolution 0 M Y 1 Y 2. ( ) Since each Y i F (R) B (R) by Fact 3.7, we have Y i for each i < 0, and similarly M. Thus, Lemma 2.8(b) implies that Eq. is Hom R (, )-exact. It follows that the complex obtained by splicing the sequences ( ) and( ) isap F - complete F -resolution of M. Thus, we have M H (F (R)). To complete the proof, use the previous two paragraphs and argue by induction on n to verify the first equality in the next sequence H n (F (R)) = H (F (R)) = GF (R) B (R). The second equality is from Proposition 6.4. Our next result contains Theorem II(a) from the introduction. orollary 6.10 If is a semidualizing R-module, then G n (GF (R) B (R)) = GF (R) B (R) for each n 1. Proof In the next sequence, the containments are from Fact 3.11 and Remark 3.13 GF (R) B (R) G n (GF (R) B (R)) = G n (H (F (R))) H n (H (F (R))) = GF (R) B (R) and the equalities are by Proposition 6.4 and Theorem 6.9.

24 426 S. Sather-Wagstaff et al. Remark 6.11 In light of orollary 6.10, it is natural to ask whether we have G(F (R)) = GF (R) B (R) for each semidualizing R-module. While Remark 3.13 and Proposition 6.4 imply that G(F (R)) GF (R) B (R), wedonot know whether the reverse containment holds. We now turn our attention to H n (F (R)) and Gn (F (R)). Proposition 6.12 Let be a semidualizing R-module and let n 1. (F (a) We have GF (R) B (R) F (R) H n (R)) GF (R) B (R). (b) If dim(r) <, thenf (R) H n (F (R)). (c) If dim(r) <, thenh n (F (R)) = GF (R) B (R) F (R). Proof (a) For the first containment, let M GF (R) B (R) F (R).SinceM B (R) F (R), Lemma 4.5(c) yields an augmented F -resolution Z 1 Z 0 M 0 that is Hom R (, )-exact; the argument of Proposition 6.4 shows that this resolution is Hom R (, F )-exact. Because M is in GF (R), Proposition 5.7 provides an augmented F -coresolution 0 M Y 1 Y 2 that is Hom R (, F )-exact. Since M B (R), the proof of Proposition 6.4 shows that this coresolution is also Hom R (, )-exact. Splicing these resolutions yields a P F -complete F -resolution of M, andsom H (F (R)) Hn (F (R)). The second containment follows from the next sequence H n (F (R)) Hn (F (R)) = GF (R) B (R) wherein the containment is by definition, and the equality is by Theorem 6.9. (b) Assume d = dim(r) <. A result of Gruson and Raynaud [20, Seconde Partie, Thm. (3.2.6)] and Jensen [19, Prop. 6] implies pd R (F) d < for each flat R-module F. We prove the result for all n 0 by induction on n. The base case n = 0 follows from Lemma 4.3. Assume n 1 and that F (R) H n 1 (F (R)). LetM H n (F (R)),andletX be a P F -complete Hn 1 (F )-resolution of M. For each i set M i = Im( i X ). This yields an isomorphism M = M 0 and, for each i, an exact sequence 0 M i+1 X i M i 0. Note that M i, X i B (R) by part (a). Let F R F (R) and let t 1. Since F (R) X i for each i, a standard dimension-shifting argument yields the first isomorphism in the next sequence Ext t R (F R, M) = Ext t+d R (F R, M d ) = Ext t+d R (F, Hom R(, M d )) = 0. The second isomorphism is a form of Hom-tensor adjointness using the fact that F is flat with the Bass class condition Ext 1 R (, M d) = 0. The vanishing follows from the inequality pd R (F) d. (c) This follows from parts (a) and (b).