Pure-Injectivity in the Category of Gorenstein Projective Modules

Transcription

1 Pure-Injectivity in the Category of Gorenstein Projective Modules Peng Yu and Zhaoyong Huang Department of Mathematics, Nanjing University, Nanjing 2193, Jiangsu Province, China Abstract In this paper, we introduce and study (weak) pure-injective Gorenstein projective modules. Let R be an artin algebra. We prove that the category of weak pure-injective Gorenstein projective left R-modules coincides with the intersection of the category of pure-injective left R-modules and that of Gorenstein projective left R-modules. As applications, we get some equivalent characterizations of virtually Gorenstein (CM-finite) algebras. Furthermore, we prove that the category of weak pure-injective Gorenstein projective left R-modules is enveloping in the category of of left R-modules; and if R is virtually Gorenstein, then it is precovering in the category of pure-injective left R- modules. 1 Introduction As a nice generalization of finitely generated projective modules over commutative noetherian local rings, Auslander and Bridger introduced in [AB] finitely generated modules of Gorenstein dimension zero; and then Enochs and Jenda generalized it in [EJ1] to Gorenstein projective modules (not necessarily finitely generated) and introduced the dual notion Gorenstein injective modules over general rings. Since then, Gorenstein projective and injective modules and related modules have become very important research objects in Gorenstein homological algebra and representation theory of algebras; see [B1, B2, BK, C1, C2, Ch, ChFH, EJ1, EJ2, Ho1, J, LY, X] and references therein. The notion of pure-injective modules (also called algebraically compact modules in [W]) was introduced by Cohn in [Co] and found many important applications in ring theory, homological algebra and representation theory of algebras. For instance, the study of generic modules (a special kind of pure-injective modules), initiated by Crawley-Boevey in [Cr], has proved to be essential in characterizing tameness of finite dimensional algebras ([K2]) and phantom maps in group algebras ([BeG]). The geometrical and topological aspects of pureinjective modules (for instance, the Ziegler spectrum and Krull-Gabriel dimension) were also widely studied in [P], [H1], [K2], and so on. In particular, in the study of the Gorenstein 2 Mathematics Subject Classification: 18G25, 16E3. Keywords: Gorenstein Projective Modules, G-Pure Exact, (Weak) Pure-Injective, Absolutely Pure, virtually Gorenstein algebras, (Pre)enveloping, (Pre)covering. address: pengyu@smail.nju.edu.cn address: huangzy@nju.edu.cn 1

2 symmetry conjecture, Beligannis introduced in [B1] the notion of virtually Gorenstein algebras as a nice generalization of Gorenstein algebras. Various equivalent characterizations of virtually Gorenstein algebras are supplied, among which the use of pure-exact structures in the category of Gorenstein projective modules attracts our interest. That is, there is a natural purity theory on this locally finitely presented category with respect to its subcategory consisting of finitely presented Gorenstein projective modules. Let R be an artin algebra. Then the category of Gorenstein projective modules is definable by [B1, Proposition 3.8]. It follows easily from [P, Theorem 3.4.8] that the pure-injective envelope of a left R-module M is Gorenstein projective if and only if M is Gorenstein projective. It indicates that the intersection of the category of pure-injective modules and that of Gorenstein projective modules provides much information on the original category. Our aim in this paper is to study the pure-injectivity in the category of Gorenstein projective modules. The paper is organized as follows. In Section 2, we give some terminology and some preliminary results. In Section 3, we introduce and study G-pure exact sequences, (weak) pure-injective, pureprojective and absolutely pure Gorenstein projective modules. Let R be a ring. We first give some equivalent characterizations of absolutely pure Gorenstein projective left R-modules in terms of the properties of G-pure exact sequences and the vanishing of Ext-funcrors in the category of contravariant functors from the category of stable finitely presented Gorentein projective left R-modules to that of abelian groups (Theorem 3.3). Then we deduce that if R is a CM-free artin algebra, then the category of Gorenstein projective left R-modules coincides with that of absolutely pure Gorenstein projective left R-modules (Corollary 3.6); and if R further is virtually Gorenstein, then these two categories coincide with the category of projective left R-modules (Corollary 3.8). This gives a partial answer to the following conjecture: For any CM-free artin algebra R, the category of Gorenstein projective left R-modules coincides with that of projective left R-modules ([C2]). Let R be an artin algebra. We prove that the category of weak pure-injective Gorenstein projective left R-modules coincides with the intersection of the category of pure-injective left R-modules and that of Gorenstein projective left R-modules (Theorem 3.1). As an application, we get that R is virtually Gorenstein if and only if any weak pure-injective Gorenstein projective left R-module is a direct limit of a family of finitely presented Gorenstein projective left R-modules (Theorem 3.15). This extends a result in [BK]. Moreover, we get that a virtually Gorenstein artin algebra R is CM-finite if and only if the category of pure-injective Gorenstein projective left R-modules coincides with that of Gorenstein projective left R-modules (Theorem 3.17). In the final of this section, we illustrate that the category of pure-injective Gorenstein projective left R-modules plays a role in that of Gorenstein projective left R-modules similar to the category of pure-injective left R-modules does in that of left R-modules (Proposition 3.19). As further applications of Theorem 3.1, we study in Section 4 the covering and enveloping properties of the category of weak pure-injective Gorenstein projective R-modules. Let R be an artin algebra. We prove that this category is enveloping in the category of left R-modules (Theorem 4.1); and if R is virtually Gorenstein, then it is precovering in the category of pure-injective left R-modules (Theorem 4.3). 2

3 2 Preliminaries In this section, we give some terminology and some preliminary results. Throughout this paper, R is an associative ring with identity, Mod R is the category of left R-modules and mod R is the category of finitely presented left R-modules. For a module M Mod R, Add M (resp. Prod M) is the subcategory of Mod R consisting of direct summands of coproducts (resp. products) of copies of M. For a subcategory C of Mod R, we use C and C to denote the stable categories of C modulo projectives and injectives, respectively. Definition 2.1 ([EJ2]) Let C D be subcategories of Mod R. The homomorphism f : C D in Mod R with C C and D D is called a C-precover of D if for any homomorphism g : C D in Mod R with C C, there exists a homomorphism h : C C such that the following diagram commutes: C h g C f D. The morphism f : C D is called right minimal if a homomorphism h : C C is an automorphism whenever f = fh. A C-precover f : C D is called a C-cover if f is right minimal. C is called (pre)covering in D if each module in D has a C-(pre)cover. Dually, the notions of a C-preenvelope, a left minimal homomorphism, a C-envelope and a (pre)enveloping subcategory are defined. Following Auslander and Reiten s terminology in [AR], a C-(pre)cover and a C-(pre)envelope are called a (minimal) right C-approximation and a (minimal) left C-approximation respectively, and a precovering subcategory and a preenveloping subcategory are called a contravariantly finite subcategory and a covariantly finite subcategory respectively. Let M Mod R and N Mod R op. Recall that a sequence in Mod R is called Hom R (, M)- exact (resp. N R -exact) if it is exact after applying the functor Hom R (, M) (resp. N R ); and for a subcategory C of Mod R, a sequence in Mod R is called Hom R (, C)- exact if it is exact after applying the functor Hom R (, M) for any M C. We use Proj R, Inj R and Flat R to denote the full subcategory of Mod R consisting of projective, injective and flat modules respectively. Definition 2.2 ([EJ1, EJ2]) (1) A module G Mod R is called Gorenstein projective if there exists a Hom R (, Proj R)- exact exact sequence: P 1 P P P 1, in Mod R with all P i and P i in Proj R and G = Im(P P ). Dually, the notion of Gorenstein injective modules is defined. (2) A module H Mod R is called Gorenstein flat if there exists an Inj R op R -exact exact sequence: F 1 F F F 1, in Mod R with all F i and F i in Flat R and H = Im(F F ). 3

4 We use GProj R, GInj R and GFlat R to denote the full subcategory of Mod R consisting of Gorenstein projective, injective and flat modules respectively. By the adjoint isomorphism theorem, it is easy to see that GProj R = GFlat R over an artin algebra R. We use Gproj R and Ginj R to denote the full subcategory of mod R consisting of Gorenstein projective and injective modules respectively. We write ( ) + := Hom Z (, Q/Z), where Z is the additive group of integers and Q is the additive group of rational numbers. Lemma 2.3 ([GoT, Lemma ]) Let A B C (2.1) be an exact sequence in Mod R. Then the following statements are equivalent. (1) M R A M R B M R C is exact for any M Mod R op. (2) M R A M R B M R C is exact for any M mod R op. (3) Hom R (N, A) Hom R (N, B) Hom R (N, C) is exact for any N mod R. (4) C + B + A + splits. (5) A B C is a direct limit of split short exact sequences. The exact sequence (2.1) is called pure exact if one of the above equivalent conditions is satisfied. Definition 2.4 ([GoT]) A submodule A of B in Mod R is called pure if the exact sequence A λ B B/A in Mod R with λ the embedding is pure exact; and a quotient module C of B in Mod R is called pure if the exact sequence Ker π B π C in Mod R with π the natural epimorphism is pure exact. We have the following easy observation. Lemma 2.5 If R is an artin algebra, then GProj R is closed under pure submodules and pure quotient modules. Proof. Let R be an artin algebra and A G B be a pure exact sequence in Mod R with G GProj R. Then B + G + A + splits with G + GInj R op by [Ho1, Theorem 3.6]. So both A + and B +, as direct summands of G +, are in GInj R op by [Ho1, Theorem 2.6]. It follows from [LY, Theorem 2.3] that both A and B are in GFlat R(= GProj R). 4

5 Definition 2.6 ([GoT]) (1) A module M Mod R is pure-projective if it is projective with respect to pure exact sequences, that is, any pure exact sequence A B M in Mod R ending at M splits. Dually, the notion of pure-injective modules is defined. (2) A module M Mod R is absolutely pure if any exact sequence in Mod R starting from M is pure exact. M B C We use PInj R and AbsR to denote the full subcategories of Mod R consisting of pureinjective modules and absolutely pure modules, respectively. The following two lemmas are used frequently in the sequel. Lemma 2.7 ([EJ2, Propositions and 5.3.9]) For any M Mod R, we have (1) M + PInj R op. (2) σ M : M M ++ via σ M (m)(f) = f(m) for any m M and f M + induces a pure exact sequence M σ M M ++ Coker σ M. Thus M is pure-injective if and only if σ M is a section. Lemma 2.8 Let R be an artin algebra. Then M ++ PInj R GProj R for any M GProj R. Proof. Let R be an artin algebra and M GProj R. Then by Lemma 2.7(1), we have M ++ PInj R. By [Ho1, Theorem 3.6] and [LY, Theorem 2.3], we have M + GInj R op and M ++ GFlat R(= GProj R). 3 Pure-injective Gorenstein projective modules In this section, we introduce and study pure-projective, (weak) pure-injective and absolutely pure Gorenstein projective modules. We begin with the following Definition 3.1 (1) An exact sequence G 1 G 2 G 3 (3.1) in Mod R with all terms in GProj R is called G-pure exact in GProj R if Hom R (G, G 1 ) Hom R (G, G 2 ) Hom R (G, G 3 ) is exact for any finitely presented Gorenstein projective left R-module G. 5

6 (2) H GProj R is called pure-projective in GProj R if Hom R (H, G 1 ) Hom R (H, G 2 ) Hom R (H, G 3 ) is an exact sequence for any G-pure exact sequence (3.1). (3) E GProj R is called pure-injective in GProj R if Hom R (G 3, E) Hom R (G 2, E) Hom R (G 1, E) is an exact sequence for any G-pure exact sequence (3.1). (4) A GProj R is called absolutely pure in GProj R if any exact sequence A G 2 G 3 in GProj R is G-pure exact. We use PP-GProj R, PI-GProj R and Abs-GProj R to denote the subcategories of GProj R consisting of pure-projective, pure-injective and absolutely pure Gorenstein projective modules, respectively. Remark 3.2 (1) Any pure exact sequence with all terms in GProj R is G-pure exact. (2) AbsR GProj R Abs-GProj R GProj R. (3) Add Gproj R PP-GProj R GProj R. (4) Proj R PI-GProj R GProj R and Proj R Abs-GProj R GProj R. For an additive category C, we use (C op, Ab) to denote the category consisting of contravariant functors from C to the category Ab of abelian groups. The following result gives some equivalent characterizations of absolutely pure Gorenstein projective modules. Theorem 3.3 The following statements are equivalent for any A GProj R. (1) A Abs-GProj R. (2) There exists a G-pure exact sequence in Mod R with P projective. A P G 1 (3) Ext 1 R (, A) = as an object of ((GprojR)op, Ab). Proof. (1) (2) It follows from the definition of Gorenstein projective modules. (2) (1) For any exact sequence A G 2 G 3 6

7 in GProj R, we can form the pushout of A P and A G 2 to get the following commutative diagram with exact columns and rows: A P G 2 G 4 G 1 G 1 G 3 G 3. Since Ext 1 R (G 3, P ) =, the middle column in the diagram splits. Applying the functor Hom R (G, ) with G Gproj R to this diagram, then the snake lemma implies that the leftmost column is G-pure exact. Hence A Abs-GProj R. (2) (3) For any G Gproj R, the given G-pure exact sequence gives rise to a long exact sequence: Hom R (G, A) Hom R (G, P ) Hom R (G, G 1 ) Ext 1 R(G, A) Ext 1 R(G, P ). Because Ext 1 R (G, P ) =, we get the equivalence between (2) and (3) easily. In the following, we give some applications of Theorem 3.3. A subcategory X of GProj R is called closed under G-pure submodules if in a G-pure exact sequence G 1 G 2 G 3, G 2 X implies G 1 X. Corollary 3.4 Abs-GProj R is closed under extensions, direct sums and G-pure submodules. Proof. It follows from Theorem 3.3 directly that Abs-GProj R is closed under extensions and G-pure submodules. Note that Ext 1 R (M, N i) = Ext 1 R (M, N i) for finitely presented module M. So Abs-GProj R of GProj R is closed under direct sums by Theorem 3.3. Corollary 3.5 For an artin algebra R, we have Gproj R Abs-GProj R = proj R. Proof. Because R is an artin algebra, for any G Gproj R Abs-GProj R there exists an exact sequence G P G 1 with P proj R and G 1 Gproj R, which splits by Theorem 3.3. Thus G proj R. Recall from [C2] that an artin algebra R is called CM-free if Gproj R = proj R. As an immediate consequence of Theorem 3.3, we have the following 7

8 Corollary 3.6 If R is a CM-free artin algebra, then GProj R = Abs-GProj R. where Recall from [B1] that an artin algebra R is called virtually Gorenstein if (GProj R) = (GInj R), (GProj R) := {Y Mod R Ext 1 R (G, Y ) = for any G GProj R} and (GInj R) := {X Mod R Ext 1 R (X, G) = for any G GInj R}. The following result gives some properties of virtually Gorenstein artin algebras. Proposition 3.7 Let R be a virtually Gorenstein artin algebra. Then we have (1) Any G-pure exact sequence is a direct limit of a system of split exact sequences in GProj R. (2) PP-GProj R = Add Gproj R. (3) PInj R GProj R PI-GProj R. (4) Abs-GProj R = Proj R. Proof. (1) Let G 1 f G2 g G3 be a G-pure exact sequence. Since R is a virtually Gorenstein artin algebra, G 3 GProj R can be written as G 3 = lim G i 3 with all Gi 3 Gproj R by [BK, Theorem 5]. Then we can form the pullback of G 2 g G3 and G i 3 α i G 3 to get the following commutative diagram with exact rows: G 1 G i g i 2 G i 3 h i β i u α i f g G 1 G 2 G 3. By the definition of G-pure exact sequences, α i factors through g, that is, there exits a homomorphism u : G i 3 G 2 such that α i = gu. Then by the universal property of pullbacks, there exists a unique homomorphism h i : G i 3 Gi 2 such that βi h i = u and g i h i = 1 G i. So the 3 upper exact sequence splits and G i 2 GProj R. By the universal property of pullbacks again, there exists a unique homomorphism γ ij : G j 2 Gi 2 for each i < j such that δ ijg j = g i γ ij and β j = β i γ ij. G 1 G j 2 g j G j 3 γ ij G 1 G i 2 β j g i δ ij G i 3 α j β i g G 1 G 2 α i G 3. 8

9 Then clearly G 2 = lim G i 2 and the assertion follows. (2) By Remark 3.2(3), it suffices to show PP-GProj R Add Gproj R. Let H PP- GProj R. Then H can be written as H = lim H i with all H i Gproj R by [BK, Theorem 5]. So the G-pure exact sequence Ker π H i π lim H i splits and H Add GProj R. It follows that PP-GProj R Add Gproj R. (3) By (1) and Lemma 2.3, any G-pure exact sequence G 1 G 2 G 3 is pure exact. Then for any E PInj R GProj R, the sequence Hom R (G 3, E) Hom R (G 2, E) Hom R (G 1, E) is exact. So E PI-GProj R and PInj R GProj R PI-GProj R. (4) Note that for an artin algebra R, Ext 1 R (, A) Gproj R = implies Ext 1 R (, A) Gproj R =. Since (Gproj R) = (GProj R) by [B2, Remark 4.6], we have Abs-GProj R = (Gproj R) GProj R = (GProj R) GProj R = Proj R by Theorem 3.3. Chen conjectured in [C2] that GProj R = Proj R if R is a CM-free artin algebra. We have the following Corollary 3.8 If R is a CM-free artin algebra, then GProj R = Proj R if and only if R is virtually Gorenstein. Proof. By [B2, Example 4.5.3], the sufficiency holds true. The necessity follows from Corollary 3.6 and Proposition 3.7(4). We introduce the notion of weak pure-injective Gorenstein projective modules as follows. Definition 3.9 A module E GProj R is called weak pure-injective in GProj R if for any pure exact sequence G 1 G 2 G 3 in GProj R, the sequence is still exact. Hom R (G 3, E) Hom R (G 2, E) Hom R (G 1, E) We use wpi-gproj R to denote the subcategory of GProj R consisting of weak pureinjective Gorenstein projective modules. Clearly, we have PI- GProj R wpi- GProj R. For an artin algebra R, we use D to denote the usual Matlis duality of R. All results in the rest of this paper are based on the following theorem. 9

10 Theorem 3.1 If R is an artin algebra, then Prod Gproj R PInj R GProj R = wpi- GProj R. Proof. By [GoT, Corollary ] and [C1, Proposition (a)], we have Prod Gproj R PInj R GProj R. Clearly, PInj R GProj R wpi-gproj R. In the following, we prove the converse inclusion. Note that PInj R = Prod mod R and R PInj R by [GoT, Corollary ]. It follows from the fact Proj R = Add R = Prod R that Proj R PInj R. Now let M wpi- GProj R( GProj R = GFlat R). Then we have a Hom R (, R)-exact and DR -exact exact sequence P := P 1 P P P 1 in Mod R with all P i and P i in Proj R such that M = Im(P P ). It gives rise to the following commutative diagram: P := P 1 P P P 1 σ P1 P ++ := P ++ 1 σ P P ++ σ P σ P 1 (P ) ++ (P 1 ) ++ L =: Coker σ P1 Coker σ P Coker σ P Coker σ P 1. All columns splits because they are pure exact by Lemma 2.7(2). By the adjoint isomorphism theorem and [R, Lemma 3.55(ii)], we have that P + is Hom R (DR, )-exact exact and P ++ is DR -exact exact. So L is also DR -exact exact in Mod R with all terms in Proj R. Then by [J, Lemma 1.7], L is Hom R (, Proj R)-exact exact and Coker σ M = Im(Coker σp Coker σ P ) GProj R. Because M wpi- GProj R, the pure exact sequence M σ M M ++ Coker σ M in Mod R with all terms in GProj R splits. By Lemma 2.8, we have M ++ PInj R GProj R. Thus, as a direct summand of M ++, M PInj R by Lemma 2.7(2). Therefore M PInj R GProj R and wpi- GProj R PInj R GProj R. As an application of Theorem 3.1, we give an equivalent characterization of pureprojective Gorenstein projective modules as follows. Corollary 3.11 Let R be an artin algebra and H GProj R. Then the following statements are equivalent. 1

11 (1) H PP-GProj R. (2) H is projective relative to any G-pure exact sequence in Mod R with E wpi-gproj R. Proof. (1) (2) It is trivial. (2) (1) For any G-pure exact sequence G 1 E G 3 f 1 f 2 G 1 G2 G3 f 2 in Mod R, we can form the pushout of G 2 G3 and G 2 commutative diagram with exact columns and rows: σ G2 G ++ 2 to get the following G 1 f 1 G 2 f 2 G 3 σ G2 G 1 G ++ 2 α D g Coker σ G2 Coker σ G2. By Lemma 2.7(2) and Theorem 3.1, we have that G 2 σ G2 G ++ 2 Coker σ G2 is a pure exact sequence in Mod R with all terms in GProj R and G ++ 2 wpi-gproj R. By [Ho1, Theorem 2.5] and the exactness of G 3 g D Coker σg2, we have D GProj R. Then it is easy to see that G 1 G ++ 2 α D is G-pure exact. Let H GProj R and u Hom R (H, G 3 ). Then there exists v Hom R (H, G ++ 2 ) such that gu = αv. Note that the above pushout diagram is also a pullback one. So there exists w Hom R (H, G 2 ) such that u = f 2 w. Thus Hom R (H, G 1 ) Hom R(H,f 1 ) Hom R (H, G 2 ) Hom R(H,f 2 ) Hom R (H, G 3 ) 11

12 is exact and H PP-GProj R. We do not know whether there exists an inclusion relation between PInj R GProj R and PI- GProj R in general. The following result shows that these two categories coincide over virtually Gorenstein artin algebras; compare it with Remark 3.2(2). Corollary 3.12 If R is a virtually Gorenstein artin algebra, then PInj R GProj R = wpi- GProj R = PI- GProj R = Prod(Ginj R op ) +. Proof. Let R be a virtually Gorenstein artin algebra. Then G-pure exact sequences and pure exact sequences coincide by Lemma 2.3 and Proposition 3.7. So PInj R GProj R = wpi-gproj R = PI-GProj R by Theorem 3.1. Because R op is also virtually Gorenstein by [B1, Theorem 8.7], we have GInj R op lim Ginj Rop by [BK, Theorem 5]. So (GProj R) + GInj R op lim Ginj R op and (Ginj R op ) + GProj R by [Ho1, Theorem 3.6]. In addition, GProj R is closed under direct summands and products by [C1, Proposition (a)]. Now it follows from [Ho2, Corollary 5.3] that PInj R GProj R = Prod(Ginj R op ) +. Recall from [B2] that an artin algebra R is called CM-finite if the number of indecomposable modules in Gproj R is finite up to isomorphism. Corollary 3.13 Let R be a CM-finite artin algebra. Then lim Gproj R wpi- GProj R. Proof. Let R be a CM-finite artin algebra. Then Gproj R = add G for some G Gproj R. So, by [KS1, Lemma 1.2] and Theorem 3.1, we have lim Gproj R = lim add G = Prod add G wpi- GProj R. In the following, we give an equivalent characterization of virtually Gorenstein artin algebras. We need the following Lemma 3.14 Let R be an artin algebra. Then lim Gproj R is closed under pure submodules. Proof. Let f 1 f 2 G 1 G2 G3 (3.2) be a pure exact sequence in Mod R with G 2 lim Gproj R. Then by [GoT, Lemma 1.2.9], we have that for any u : M G 1 with M mod R, f 1 u factors through some G Gproj R, that is, there exist s : M G and w : G G 2 such that f 1 u = ws. Consider the following commutative diagram with exact rows: M s G t N u G 1 f 1 G 2 f 2 G 3, w v 12

13 where N = Coker s and v is an induced homomorphism. Since the bottom row in this diagram (that is, (3.2)) is pure exact by (1), there exists h : N G 2 such that v = f 2 h. Notice that f 2 (w ht) = f 2 w f 2 ht = f 2 w vt =. So w ht factors through f 1, that is, there exists k : G G 1 such that w ht = f 1 k. Since f 1 is monic and f 1 (ks u) = f 1 ks f 1 u = (w ht)s f 1 u = ws hts f 1 u = ws f 1 u =, we have ks u = and ks = u, that is, u factors through G( Gproj R). Thus G 1 lim Gproj R by [GoT, Lemma 1.2.9]. Beligiannis and Krause proved in [BK, Theorem 5] that an artin algebra R is virtually Gorenstein if and only if GProj R lim Gproj R. The following theorem extends this result. Theorem 3.15 Let R be an artin algebra. Then the following statements are equivalent. (1) R is virtually Gorenstein. (2) wpi-gproj R lim Gproj R. Proof. (1) (2) By [BK, Theorem 5]. (2) (1) Let G GProj R. Consider the pure exact sequence: G σ G G ++ Coker σ G. By Lemma 2.8 and Theorem 3.1, we have G ++ PInj R GProj R = wpi-gproj R. So G ++ lim Gproj R by (2), and hence G lim Gproj R by Lemma It follows from [BK, Theorem 5] that R is virtually Gorenstein. By Corollary 3.13 and Theorem 3.15, we have the following Corollary 3.16 Let R be a virtually Gorenstein CM-finite artin algebra. Then wpi- GProj R = lim Gproj R. Recall from [K2] that a subcategory of Mod R is called definable if it is closed under direct limits, direct products and pure submodules in Mod R. The following result gives an equivalent characterization for virtually Gorenstein artin algebras being CM-finite. It induces that both PI-GProj R and wpi-gproj R are proper subcategories of GProj R in general. Theorem 3.17 Let R be a virtually Gorenstein artin algebra. Then the following statements are equivalent. (1) R is CM-finite. (2) PI-GProj R = GProj R. Proof. Let R be a virtually Gorenstein artin algebra. Then GProj R is definable by [B1, Proposition 3.8]. It follows from [B2, Theorem 4.1] and [K2, Corollary 2.7] that R is CM-finite if and only if GProj R PInj R. Now the assertion follows from Corollary

14 By [B1, Theorem 3.5], we have that GProj R is covering and enveloping in Mod R. For a module M Mod R, we use α M : G M M and α M : M G M to denote the GProj R-cover and the GProj R-envelope of M, respectively. By Theorem 3.1, we also have the following Proposition 3.18 Let R be an artin algebra. have If PI-GProj R = wpi-gproj R, then we (1) Any G-pure exact sequence is pure exact. (2) lim Gproj R is closed under G-pure submodules. Proof. (1) Let f 1 f 2 G 1 G2 G3 (3.3) be a G-pure exact sequence in Mod R with all terms in GProj R. Let M Mod R be pure-injective. By [KS2, Lemma 2.2], both G M and Ker α M are pure-injective. So G M PInj R GProj R = wpi-gproj R = PI-GProj R by Theorem 3.1 and assumption. Consider the following commutative diagram with exact columns and rows: Hom R (G 3, Ker α M ) Hom R (G 2, Ker α M ) Hom R (G 1, Ker α M ) Hom R (G 3, G M ) Hom R (G 2, G M ) Hom R (G 1, G M ) Then Hom R (G 3, M) Hom R(f 2,M) Hom R (G 2, M) Hom R(f 1,M) Hom R (G 1, M). Hom R (G 3, M) Hom R(f 2,M) Hom R (G 2, M) Hom R(f 1,M) Hom R (G 1, M) is exact. Let N mod R op. Then N + PInj R by Lemma 2.7(1). Note that for any 1 i 3, we have (N G i ) + = HomR (G i, N + ) by the adjoint isomorphism theorem. By the above argument, we have an exact sequence: Hom R (G 3, N + ) Hom R(f 2,N + ) Hom R (G 2, N + ) Hom R(f 1,N + ) Hom R (G 1, N + ), 14

15 which induces the following exact sequence: N f 1 N f 2 N R G 1 N R G 2 N R G 3. Thus (3.3) is pure-exact. (2) By (1) and Lemma By Corollary 3.12, if R is a virtually Gorenstein CM-finite artin algebra, then the assumption in Proposition 3.18 is satisfied. For a subcategory C of Mod R and an object M C, the stable objects of M in C and C are denoted by M and M, respectively. If R is an artin algebra, then GProjR is a compactly generated triangulated category by [Bu, Remark 4.8] and [B1, Theorem 6.6]. Moreover, an artin algebra R is virtually Gorenstein if and only if GprojR coincides with the compact subcategory of GProjR by [B1, Theorem 8.2]. For a compactly generated triangulated category T, there is a theory of purity which parallels that of module categories ([K1]). A triangle in T is called pure if for any compact object X in T, A B C A[1] (3.4) T (X, A) T (X, B) T (X, C) is exact. An object E T is called pure-injective if for any pure triangle as (3.4), T (C, E) T (B, E) T (A, E) is exact. Let A be an abelian category. We use D + (A) to denote the bounded below derived category of A. Let Inj A be the full subcategory of A consisting of injective objects, we use K + (Inj A) to denote the category whose objects are bounded below complexes of injective objects and whose morphisms are morphisms of complexes modulo homotopic equivalence. It was proved in [H2, Theorem 23] that the functor Ext 1 R(,?) : PInjR ((modr) op, Ab) via N Ext 1 R(, N) yields an equivalence between PInjR and the subcategory of injective objects of ((modr) op, Ab). Then by [GM, Theorem ], there exists the following equivalence of triangulated categories: K + (PInjR) D + ((modr) op, Ab). Put Hom R (X, Y ) := Hom R (X, Y )/{the morphism X Y factoring through some projective left R-module}. We use PI-GProjR to denote the full subcategory of GProjR consisting of pure-injective objects. We have the following comparable result for a virtually Gorenstein artin algebra R, which means that PI- GProj R plays a role in GProj R similar to PInj R does in Mod R. 15

16 Proposition 3.19 Let R be a virtually Gorenstein artin algebra. Then the following statements are equivalent. (1) M PI-GProj R. (2) M is pure-injective in GProjR. (3) Hom R (, M)( = Ext 1 R (, ΩM)) is injective in ((GprojR)op, Ab), where ΩM is the first syzygy of M. Consequently, there exists the following equivalence of triangulated categories: K + (PI- GProjR) D + ((GprojR) op, Ab). Proof. (1) (2) follows from Corollary 3.12 and [B1, Lemma 6.4]. (2) (3) follows from [K1, Corollary 1.9]. Now we have and hence we have PI- GProjR = PI-GProjR (by the equivalence between (1) and (2)) Inj((GprojR) op, Ab) (by [K1, Corollary 1.9]), K + (PI- GProjR) = K + (PI-GProjR) K + (Inj((GprojR) op, Ab)) 4 Envelopes and Covers D + ((GprojR) op, Ab) (by [GM, Theorem ]). As further applications of Theorem 3.1, we study in this section the covering and enveloping properties of wpi-gproj R. It should be remarked that all results obtained here rely on the fact that GProj R is covering and enveloping in Mod R ([B1, Theorem 3.5]). We first prove the following Theorem 4.1 If R is an artin algebra, then wpi-gproj R is enveloping in Mod R. Proof. Let M Mod R and v : M X be in Mod R with X wpi-gproj R( GProj R). We claim that σ G M α M : M (G M ) ++ is a wpi-gproj R-preenvelope of M. Consider the following diagram: G M σ G M (G M ) ++ Coker σ G M α M M v X. 16

17 Then there exists δ : G M X such that v = δα M. Because X PInj R and the upper row in the above diagram is pure-exact by Theorem 3.1 and Lemma 2.7(2) respectively, there exists γ : (G M ) ++ X such that δ = γσ G M. So v = δα M = γ(σ G M α M ) and σ G M α M is a wpi-gproj R-preenvelope of M. The claim follows. Moreover, note that wpi-gproj R is closed under direct summands. Then from [K2, Theorem 3.14] we conclude that wpi-gproj R is enveloping in Mod R. To give the covering property of wpi-gproj R, we need the following Lemma 4.2 If R is a virtually Gorenstein artin algebra, then the GProj R-cover of any finitely generated R-module is finitely generated. Proof. By [B1, Corollary 8.3], R is virtually Gorenstein if and only if the GProj R-cover of R/r is finitely generated, where r is the Jacobson radical of R. Since R/r is a semisimple R-module containing each simple R-module as a summand, it follows that the GProj R-cover of any simple R-module is finitely generated by assumption. Let M mod R and let α K G M M M be an exact sequence in Mod R with α M the GProj R-cover of M. We will prove that G M is finitely generated by using induction on the length l(m) of M. Let l(m) = n. If n = 1, then the assertion holds true trivially. Now suppose n 2. Let M 1 be a maximal submodule of M. Then we have an exact sequence M 1 M π S in mod R with S(= M/M 1 ) simple. By the Wakamatsu lemma (c.f. [EJ2, Corollary 7.2.3]), we have Ker α M1 (GProj R). Then it is easy to get Ext 1 R(G S, G M1 ) = Ext 1 R(G S, M 1 ). (4.1) Now we construct the following commutative diagram with exact rows as follows. First take the pullback of π and α S to get the lower commutative diagram, and then the isomorphism (4.1) gives rise to the upper commutative diagram. G M1 G G S α M1 M 1 L g G S f α S M 1 M π S. Since both Ker α M1 and Ker α S are in (GProj R) by the Wakamatsu lemma again, so is Ker fg. Then the epimorphism fg : G M is a GProj R-precover of M. Because both G M1 and G S are finitely generated by the induction hypothesis, so is G. Therefore, the GProj R-cover of M, as a summand of G, is also finitely generated. Now we are in a position to prove the following 17

18 Theorem 4.3 If R is a virtually Gorenstein artin algebra, then wpi-gproj R is precovering in PInj R. Proof. Because PInj R = Prod mod R, it suffices to show that any M = i I M i with M i mod R has a wpi-gproj R-precover by [EJ2, p.16, Exercise 3]. Put α := i I α M i. Since R is virtually Gorenstein, each G Mi is finitely generated by Lemma 4.2. So i I G M i Prod Gproj R wpi-gproj R by Theorem 3.1. We claim that α is a wpi-gproj R-precover of M. Let u : E M in Mod R with E wpi-gproj R( GProj R). Consider the following commutative diagram: i I G M i α p i E u i I M i π i G Mi α Mi M i, where for any i I, p i and π i are the i-th projections of i I G M i and i I M i respectively. Then there exists β : E G Mi such that π i u = α Mi β. By the universal property of direct products, there exists γ : E i I G M i such that β = p i γ. So we have π i u = α Mi β = α Mi p i γ = π i αγ for any i I. It induces that u = αγ and the claim follows. Acknowledgments. This research was partially supported by NSFC (Grant No ) and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. References [AB] M. Auslander and M. Bridger, Stabe Module Theory, Memoirs Amer. Math. Soc. 94, Amer. Math. Soc., Providence, RI, [AR] M. Auslander and I. Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991) [B1] A. Beligiannis, Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras, J. Algebra 288 (25), [B2] A. Beligiannis, On algebras of finite Cohen-Macaulay type, Adv. Math. 226 (211), [BK] A. Beligiannis and H. Krause, Thick subcategories and virtually Gorenstein algebras, Illinois J. Math. 52 (28), [BeG] D.J. Benson and G.Ph. Gnacadja, Phantom maps and purity in modular representation theory I, Algebraic topology (Kazimierz Dolny, 1997), Fund. Math. 161 (1999),

19 [Bu] R.O. Buchweitz, Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, Unpublished manuscript, [C1] X.W. Chen, Gorenstein Homological Algebra of Artin Algebras, Postdoctoral Report, USTC, 21, Preprint is available at xwchen/. [C2] X.W. Chen, Algebras with radical square zero are either self-injective or CM-free, Proc. Amer. Math. Soc. 14 (212), [Ch] L.W. Christensen, Gorenstein Dimensions, Lecture Notes in Math. 1747, Springer- Verlag, Berlin, 2. [ChFH] L.W. Christensen, H-B. Foxby and H. Holm, Beyond totally reflexive modules and back: a survey on Gorenstein dimensions, in: Commutative algebra Noetherian and non-noetherian perspectives, Springer, New York, 211, pp [Co] P.M. Cohn, On the free product of associative rings, Math. Z. 71 (1959), [Cr] W. Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. Soc. 63 (1991), [EJ1] E.E. Enochs and O.M.G. Jenda, Gorenstein injective and projective modules, Math. Z. 22 (1995), [EJ2] E.E. Enochs and O.M.G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Math. 3, Walter de Gruyter, Berlin, 2. [GM] S.I. Gelfand and Y.I. Manin, Methods of Homological Algebra, Second edition, Springer Monographs in Math., Springer-Verlag, Berlin, 23. [GoT] R. Göbel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, De Gruyter Expositions in Math. 41, Walter de Gruyter GmbH & Co. KG, Berlin, 26. [H1] I. Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, Proc. London Math. Soc. 74 (1997), [H2] I. Herzog, Contravariant functors on the category of finitely presented modules, Israel J. Math. 167 (28), [Ho1] H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (24), [Ho2] H. Holm, Modules with cosupport and injective functors, Algebr. Represent. Theory 13 (21), [J] P. Jøgensen, Gorenstein projective resolutions and Tate cohomology, J. Eur. Math. Soc. 9 (27), [K1] H. Krause, Smashing subcategories and the telescope conjecture an algebraic approach, Invent. Math. 139 (2),

20 [K2] H. Krause, The Spectrum of a Module Category, Mem. Amer. Math. Soc. (77) 149, 21. [KS1] H. Krause and Ø. Solberg, Applications of cotorsion pairs, J. London Math. Soc. 68 (23), [KS2] H. Krause and Ø. Solberg, Filtering modules of finite projective dimension, Forum Math. 15 (23), [LY] Z.K. Liu and X.Y. Yang, Gorenstein projective, injective and flat modules, J. Aust. Math. Soc. 87 (29), [P] [R] M. Prest, Purity, Spectra and Localisation, Encyclopedia of Mathematics and its Applications 121, Cambridge Univ. Press, Cambridge, 29. J.J. Rotman, An Introduction to Homological Algebra, Second Edition, Universitext, Springer, New York, 29. [W] R.B.Jr. Warfield, Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), [X] J.Z. Xu, Flat Covers of Modules, Lecture Notes in Math. 1634, Springer-Verlag, Berlin,