An Auslander-type result for Gorenstein-projective modules
aa r X i v : . [ m a t h . R T ] A p r AN AUSLANDER-TYPE RESULT FORGORENSTEIN-PROJECTIVE MODULES
XIAO-WU CHEN
Department of MathematicsUniversity of Science and Technology of ChinaHefei 230026, P. R. China
Abstract.
An artin algebra A is said to be CM-finite if there are only finitelymany, up to isomorphisms, indecomposable finitely generated Gorenstein-projective A -modules. We prove that for a Gorenstein artin algebra, it is CM-finite if andonly if every its Gorenstein-projective module is a direct sum of finitely generatedGorenstein-projective modules. This is an analogue of Auslander’s theorem onalgebras of finite representation type ([3, 4]). Introduction
Let A be an artin R -algebra, where R is a commutative artinian ring. Denoteby A -Mod (resp. A -mod) the category of (resp. finitely generated) left A -modules.Denote by A -Proj (resp. A -proj) the category of (resp. finitely generated) projective A -modules. Following [21], a chain complex P • of projective A -modules is definedto be totally-acyclic , if for every projective module Q ∈ A -Proj the Hom-complexesHom A ( Q, P • ) and Hom A ( P • , Q ) are exact. A module M is said to be Gorenstein-projective if there exists a totally-acyclic complex P • such that the 0-th cocycle Z ( P • ) = M . Denote by A -GProj the full subcategory of Gorenstein-projectivemodules. Similarly, we define finitely generated Gorenstein-projective modules byreplacing all modules above by finitely generated ones, and we also get the category A -Gproj of finitely generated Gorenstein-projective modules [17]. It is known that A -Gproj = A -GProj ∩ A -mod ([14], Lemma 3.4). Finitely generated Gorenstein-projective modules are also referred as maximal Cohen-Macaulay modules. Thesemodules play a central role in the theory of singularity [11, 12, 10, 14] and of relativehomological algebra [9, 17].An artin algebra A is said to be CM-finite if there are only finitely many, upto isomorphisms, indecomposable finitely generated Gorenstein-projective modules.Recall that an artin algebra A is said to be of finite representation type if there areonly finitely many isomorphism classes of indecomposable finitely generated modules.Clearly, finite representation type implies CM-finite. The converse is not true, ingeneral.Let us recall the following famous result of Auslander [3, 4] (see also Ringel-Tachikawa [26], Corollary 4.4) : This project was supported by China Postdoctoral Science Foundation No. 20070420125, and wasalso partially supported by the National Natural Science Foundation of China (Grant No.s 10725104,10501041 and 10601052).E-mail: [email protected].
Auslander’s Theorem
An artin algebra A is of finite representation type if andonly if every A -module is a direct sum of finitely generated modules, that is, A is leftpure semisimple, see [31].Inspired by the theorem above, one may conjecture the following Auslander-typeresult for Gorenstein-projective modules: an artin algebra A is CM-finite if and onlyif every Gorenstein-projective A -module is a direct sum of finitely generated ones.However we can only prove this conjecture in a nice case.Recall that an artin algebra A is said to be Gorenstein [19] if the regular module A has finite injective dimension both at the left and right sides. Our main result is Main Theorem
Let A be a Gorenstein artin algebra. Then A is CM-finite if andonly if every Gorenstien-projective A -module is a direct sum of finitely generatedGorenstein-projective modules.Note that our main result has a similar character to a result by Beligiannis ([9],Proposition 11.23), and also note that similar concepts were introduced and thensimilar results and ideas were developed by Rump in a series of papers [28, 29, 30].2. Proof of Main Theorem
Before giving the proof, we recall some notions and known results.2.1. Let A be an artin R -algebra. By a subcategory X of A -mod, we mean a fulladditive subcategory which is closed under taking direct summands. Let M ∈ A -mod.We recall from [8, 6] that a right X -approximation of M is a morphism f : X −→ M such that X ∈ X and every morphism from an object in X to M factors through f . The subcategory X is said to be contravariantly-finite in A -mod if each finitelygenerated modules has a right X -approximation. Dually, one defines the notions of left X -approximations and covariantly-finite subcategories. The subcategory X issaid to be functorially-finite in A -mod if it is contravariantly-finite and covariantly-finite. Recall that a morphism f : X −→ M is said to be right minimal , if for eachendomorphism h : X −→ X such that f = f ◦ h , then h is an isomorphism. A right X -approximation f : X −→ M is said to be a right minimal X -approximation if it isright minimal. Note that if a right approximation exists, so does right minimal ones;a right minimal approximation, if in existence, is unique up to isomorphisms. Fordetails, see [8, 6, 7].The following fact is known. Lemma 2.1.
Let A be an artin algebra. Then(1). The subcategory A -Gproj of A -mod is closed under taking direct summands,kernels of epimorphisms and extensions, and contains A -proj .(2). The category A -Gproj is a Frobenius exact category [22] , whose relativeprojective-injective objects are precisely contained in A -proj . Thus the stable cate-gory A -Gproj modulo projectives is a triangulated category.(3). Let A be Gorenstein. Then the subcategory A -Gproj of A -mod is functorially-finite.(4). Let A be Gorenstein. Denote by { S i } ni =1 a complete list of pairwise noniso-morphic simple A -modules. Denote by f i : X i −→ S i the right minimal A -Gproj -approximations. Then every finitely generated Gorenstein-projective module M is a N AUSLANDER-TYPE RESULT FOR GORENSTEIN-PROJECTIVE MODULES 3 direct summand of some module M ′ , such that there exists a finite chain of submod-ules M ⊆ M ⊆ · · · ⊆ M m − ⊆ M m = M ′ with each subquotient M j /M j − lyingin { X i } ni =1 . Proof.
Note that A -Gproj is nothing but X ω with ω = A -proj in [6], section 5.Thus (1) follows from [6], Proposition 5.1, and (3) follows from [6], Corollary 5.10(1)(just note that in this case, A A is a cotilting module).Since A -Gproj is closed under extensions, thus it becomes an exact category in thesense of [22]. The property of being Frobenius and the characterization of projective-injective objects follow directly from the definition, also see [14], Proposition 3.1(1).Thus by [18], chapter 1, section 2, the stable category A -Gproj is triangulated.By (1) and (3), we see that (4) is a special case of [6], Proposition 3.8. (cid:4) Let R be a commutative artinian ring as above. An additive category C is saidbe to R -linear if all its Hom-spaces are R -modules, and the composition maps are R -bilinear. An R -linear category is said to be hom-finite , if all its Hom-spaces arefinitely generated R -modules. Recall that an R -variety C means a hom-finite R -linearcategory which is skeletally-small and idempotent-split (that is, for each idempotentmorphism e : X −→ X in C , there exists u : X −→ Y and v : Y −→ X such that e = v ◦ u and Id Y = u ◦ v ). It is well-known that a skeletally-small R -linear categoryis an R -variety if and only if it is hom-finite and Krull-Schmidt (i.e., every objectis a finite sum of indecomposable objects with local endomorphism rings). See [27],p.52 or [15], Appendix A. Then it follows that any factor category ([7], p.101) of an R -variety is still an R -variety.Let C be an R -variety. We will abbreviate the Hom-space Hom C ( X, Y ) as (
X, Y ).Denote by ( C op , R -Mod) (resp. ( C op , R -mod)) the category of contravariant R -linearfunctors from C to R -Mod (resp. R -mod). Then ( C op , R -Mod) is an abelian categoryand ( C op , R -mod) is its abelian subcategory. Denote by ( − , X ) the representablefunctor for each X ∈ C . A functor F is said to be finitely generated if there exists anepimorphism ( − , C ) −→ F for some object C ∈ C ; F is said to be finitely presented (=coherent) [2, 3], if there exists an exact sequence of functors ( − , C ) −→ ( − , C ) −→ F −→ . Denote by fp ( C ) the subcategory of ( C op , R -Mod) consisting of finitelypresented functors. Clearly, fp ( C ) ⊆ ( C op , R -mod). Recall the duality D = Hom R ( − , E ) : R -mod −→ R -mod , where E is injective hull of R/ rad( R ) as an R -module. Therefore, it induces duality D : ( C op , R -mod) −→ ( C , R -mod) and D : ( C , R -mod) −→ ( C op , R -mod). The R -variety C is called a dualizing R -variety [5], if this duality preserves finitely presentedfunctors.The following observation is important. Lemma 2.2.
Let A be a Gorenstein artin R -algebra. Then the stable category A -Gproj is a dualizing R -variety. Proof.
Since A -Gproj ⊆ A -mod is closed under taking direct summands, thusidempotents-split. Therefore, we infer that A -Gproj is an R -variety, and its stablecategory A -Gproj is also an R -variety. By Lemma 2.1(3), the subcategory A -Gprojis functorially-finite in A -mod, then by a result of Auslander-Smalø ([8], Theo-rem 2.4(b)) A -Gproj has almost-split sequences, and thus theses sequences induce XIAO-WU CHEN
Auslander-Reiten triangles in A -Gproj (Let us remark that it is Happel ([19], 4.7)who realized this fact for the first time). Hence the triangulated category A -Gprojhas Auslander-Reiten triangles, and by a theorem of Reiten-Van den Bergh ([25],Theorem I.2.4) we infer that A -Gproj has Serre duality. Now by [20], Proposition2.11 (or [13], Corollary 2.6), we deduce that A -Gproj is a dualizing R -variety. Let usremark that the last two cited results are given in the case where R is a field, howeverone just notes that the results can be extended to the case where R is a commutativeartinian ring without any difficulty. (cid:4) For the next result, we recall more notions on functors over varieties. Let C be an R -variety and let F ∈ ( C op , R -Mod) be a functor. Denote by ind( C ) the complete setof pairwise nonisomorphic indecomposable objects in C . The support of F is definedto supp( F ) = { C ∈ ind( C ) | F ( C ) = 0 } . The functor F is simple if it has no nonzeroproper subfunctors, and F has finite length if it is a finite iterated extension of simplefunctors. Observe that F has finite length if and only if F lies in ( C op , R -mod) andsupp( F ) is a finite set. The functor F is said to be noetherian , if its every subfunctoris finitely generated. It is a good exercise to show that a functor is noetherian if andonly if every ascending chain of subfunctors in F becomes stable after finite steps (onemay use the fact: for a finitely generated functor F with epimorphism ( − , C ) −→ F ,then for any subfunctor F ′ of F , F ′ = F provided that F ′ ( C ) = F ( C )). Observethat a functor having finite length is necessarily noetherian by an argument on itstotal length (i.e., l ( F ) = P C ∈ ind( C ) l R ( F ( C )), where l R denotes the length functionon finitely generated R -modules).The following result is essentially due to Auslander (compare [3], Proposition 3.10). Lemma 2.3.
Let C be a dualizing R -variety, F ∈ ( C op , R -mod) . Then F has finitelength if and only if F is finitely presented and noetherian. Proof.
Recall from [5], Corollary 3.3 that for a dualizing R -variety, functors havingfinite length are finitely presented. So the “only if” follows.For the “if” part, assume that F is finitely presented and noetherian. Since F isfinitely presented, by [5], p.324, we have the filtration of subfunctors0 = soc ( F ) ⊆ soc ( F ) ⊆ · · · ⊆ soc i +1 ( F ) ⊆ · · · where soc ( F ) is the socle of F , and in general soc i +1 is the preimage of the socle of F/ soc i ( F ) under the canonical morphism F −→ F/ soc i ( F ). Since F is noetherian,we get soc i F = soc i +1 ( F ) for some i , and that is, the socle of F/ soc i ( F ) is zero.However, by the dual of [5], Proposition 3.5, we know that for each nonzero finitelypresented functor F , the socle soc( F ) is necessarily nonzero and finitely generatedsemisimple. In particular, soc( F ) has finite length, and thus it is finitely presented.Note that fp ( C ) ⊆ ( C op , R -mod) is an abelian subcategory, closed under extensions.Thus F/ soc ( F ) is finitely presented. Applying the above argument to F/ soc ( F ),we obtain that soc ( F ), as the extension between the socles of two finitely presentedfunctors, has finite length. In general, one proves that F/ soc i ( F ) is finitely presentedand soc i +1 ( F ) has finite length for all i . Hence soc( F/ soc i ( F )) = 0 will imply that F/ soc i ( F ) = 0, i.e., F = soc i ( F ), which has finite length. (cid:4) Let us consider the category A -GProj. Similar to Lemma 2.1(1),(2), we recallthat A -GProj ⊆ A -Mod is closed under taking direct summands, kernels of epi-morphisms and extensions, and it is a Frobenius exact category with (relative) N AUSLANDER-TYPE RESULT FOR GORENSTEIN-PROJECTIVE MODULES 5 projective-injective objects precisely contained in A -Proj. Consider the stable cate-gory A -GProj, which is also triangulated and has arbitrary coproducts. Recall that inan additive category T with arbitrary coproducts, an object T is said to be compact ,it the functor Hom T ( T, − ) commutes with coproducts. Denote the full subcategoryof compact objects by T c . If we assume further that T is triangulated, then T c is athick triangulated subcategory. We say that T is a compactly generated [23, 24], ifthe subcategory T c is skeletally-small and for each object X , X ≃ T ( T, X ) = 0 for every compact object T .Note that in our situation, we always have an inclusion A -Gproj ֒ → A -GProj,and in fact, we view it as A -Gproj ⊆ ( A -GProj) c . Next lemma, probably known toexperts, states the converse in Gorenstein case. It is a special case of [14], Theorem4.1 (compare [10], Theorem 6.6). One may note that in the artin case, the category A -Gproj is idempotent-split. Lemma 2.4.
Let A be an Gorenstein artin algebra. Then the triangulated category A -GProj is compactly generated and A -Gproj ⊆ ( A -GProj) c is dense (i.e., surjectiveup to isomorphisms). Proof of Main Theorem:
Assume that A is an artin R -algebra. Set C = A -Gproj, by Lemma 2.2, C is a dualizing R -variety. For a finitely generated Gorenstein-projective module M , we will denote by ( − , M ) the functor Hom C ( − , M ); for an ar-bitrary module X , we denote by ( − , X ) | C the restriction of the functor Hom A ( − , X )to C .For the “if” part, we assume that each Gorenstein-projective module is a directsum of finitely generated ones. It suffices to show that the set ind( C ) is finite. Forthis end, assume that M is a finitely generated Gorenstein-projective module. Weclaim that the functor ( − , M ) is noetherian. In fact, given a subfunctor F ⊆ ( − , M ),first of all, we may find an epimorphism ⊕ i ∈ I ( − , M i ) −→ F, where each M i ∈ C and I is an index set. Compose this epimorphism with theinclusion of F into ( − , M ), we get a morphism from ⊕ i ∈ I ( − , M i ) to ( − , M ). By theuniversal property of coproducts and then by Yoneda’s Lemma, we have, for each i ,a morphism θ i : M i −→ M , such that F is the image of the morphism X i ∈ I ( − , θ i ) : ⊕ i ∈ I ( − , M i ) −→ ( − , M ) . Note that ⊕ i ∈ I ( − , M i ) ≃ ( − , ⊕ i ∈ I M i ) | C , and the morphism above is also induced bythe morphism P i ∈ I θ i : ⊕ i ∈ I M i −→ M . Form a triangle in A -GProj K [ − −→ ⊕ i ∈ I M i P i ∈ I θ i −→ M φ −→ K. By assumption, we have a decomposition K = ⊕ j ∈ J K j where each K j is finitelygenerated Gorenstein-projective. Since the module M is finitely generated, we inferthat φ factors through a finite sum ⊕ j ∈ J ′ K j , where J ′ ⊆ J is a finite subset. In otherwords, φ is a direct sum of M φ ′ −→ ⊕ j ∈ J ′ K j and 0 −→ ⊕ j ∈ J \ J ′ K j . XIAO-WU CHEN
By the additivity of triangles, we deduce that there exists a commutative diagram ⊕ i ∈ I M i P i ∈ I θ i MM ′ ⊕ ( ⊕ j ∈ J \ J ′ K j )[ − ( θ ′ , M where the left side vertical map is an isomorphism, and M ′ and θ ′ are given by thetriangle ( ⊕ j ∈ J ′ K j )[ − −→ M ′ θ ′ −→ M φ ′ −→ ⊕ j ∈ J K j . Note that M ′ ∈ C , and by theabove diagram we infer that F is the image of the morphism ( − , θ ′ ) : ( − , M ′ ) −→ ( − , M ), and thus F is finitely-generated. This proves the claim.By the claim, and by Lemma 2.3, we deduce that for each M ∈ C , the func-tor ( − , M ) has finite length, in particular, supp(( − , M )) is finite. Assume that { S i } ni =1 is a complete list of pairwise nonisomorphic simple A -modules. Denoteby f i : X i −→ S i the right minimal A -Gproj-approximations. By Lemma 2.1(4),the module M is a direct summand of M ′ and we have a finite chain of sub-modules of M ′ with factors being among X i ’s. Then it is not hard to see thatsupp(( − , M )) ⊆ supp(( − , M ′ )) ⊆ S ni =1 supp(( − , X i )) for every M ∈ C . Therefore wededuce that ind( C ) = S ni =1 supp(( − , X i )), which is finite.For the “only if” part, assume that A is a CM-finite Gorenstein algebra. Thenthe set ind( C ) is finite, say ind( C ) = { G , G , · · · , G m } . Let B = End C ( ⊕ mi =1 G i ) op .Then B is also an artin R -algebra. Note that for each C ∈ C , the Hom-spaceHom C ( ⊕ mi =1 G i , C ) has a natural left B -module structure, moreover, it is a finitelygenerated projective B -module. In fact, we get an equivalence of categoriesΦ = Hom C ( ⊕ mi =1 G i , − ) : C −→ B -proj . Then the equivalence above naturally induces the following equivalences, still denotedby Φ Φ : fp ( C ) −→ B -mod , Φ : ( C op , R -Mod) −→ B -Mod . In what follows, we will use these equivalences. By [24], p.169 (or [13], Proposition2.4), we know that the category fp ( C ) is a Frobenius category. Therefore, via Φ,we get that B is a self-injective algebra. Therefore by [1], Theorem 31.9, we getthat B -Mod is also a Frobenius category, and by [1], p.319, every projective-injective B -module is of form ⊕ mi =1 Q ( I i ) m , where { Q , Q , · · · , Q m } is a complete set of inde-composable projective B -modules such that Q i = Φ( G i ), and each I i is some indexset, and Q ( I i ) i is the corresponding coproduct.Take { P , P , · · · , P n } to be a complete set of pairwise nonisomorphic indecom-posable projective A -modules. Let G ∈ A -GProj. We will show that G is a directsum of some copies of G i ’s and P j ’s. Then we are done. Consider the functor( − , G ) | C , which is cohomological, and thus by [13], Lemma 2.3 (or [24], p.258), weget Ext ( F, ( − , G ) | C ) = 0 for each F ∈ fp ( C ), where the Ext group is taken in( C op , R -Mod). Via Φ and applying the Baer’s criterion, we get that ( − , G ) | C is aninjective object, and thus by the above, we get an isomorphism of functors ⊕ mi =1 ( − , T i ) ( I i ) −→ ( − , G ) | C , where I i are some index sets. As in the first part of the proof, we get a morphism θ : ⊕ mi =1 T ( I i ) i −→ T such that it induces the isomorphism above. Form the triangle N AUSLANDER-TYPE RESULT FOR GORENSTEIN-PROJECTIVE MODULES 7 in A -GProj ⊕ mi =1 G ( I i ) i θ −→ T −→ X −→ ( ⊕ mi =1 G ( I i ) i )[1] . For each C ∈ C , applying the cohomological functor Hom A - GProj ( C, − ) and by theproperty of θ , we obtain thatHom A - GProj ( C, X ) = 0 , ∀ C ∈ C . By Lemma 2.4, the category A -GProj is generated by C , and thus X ≃
0, and hence θ is an isomorphism in the stable category A -GProj. Thus it is well-known (say, by[16], Lemma 1.1) that this will force an isomorphism in the module category ⊕ mi =1 G ( I i ) i ⊕ P ≃ G ⊕ Q, where P and Q are projective A -modules. Now by [1], p.319, again, P is a directsum of copies of P j ’s. Hence the combination of Azumaya’s Theorem and Crawlay-Jønsson-Warfield’s Theorem ([1], Corollary 26.6) applies in our situation, and thus weinfer that G is isomorphic to a direct sum of copies of G i ’s and P j ’s. This completesthe proof. (cid:4) Acknowledgement:
The author would like to thank the referee very much forhis/her helpful suggestions and comments.
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