aa r X i v : . [ m a t h . R A ] S e p ON EXTENSIONS OF COVARIANTLY FINITE SUBCATEGORIES
XIAO-WU CHEN
Department of MathematicsUniversity of Science and Technology of ChinaHefei 230026, P. R. China
Abstract.
In [6], Gentle and Todorov proved that in an abelian category withenough projective objects, the extension subcategory of two covariantly finite sub-categories is still covariantly finite. We give an counterexample to show thatGentle-Todorov’s theorem may fail in arbitrary abelian categories; we also provethat a triangulated version of Gentle-Todorov’s theorem holds; we make applica-tions of Gentle-Todorov’s theorem to obtain short proofs to a classical result byRingel and a recent result by Krause and Solberg. Main Theorems
Let C be an additive category. By a subcategory X of C we always mean a fulladditive subcategory. Let X be a subcategory of C and let M ∈ C . A morphism x M : M −→ X M is called a left X -approximation of M if X M ∈ X and everymorphism from M to an object in X factors through x M . The subcategory X is saidto be covariantly finite in C , if every object in C has a left X -approximation. Thenotions of left X -approximation and covariantly finite are also known as X -preenvelop and preenveloping , respectively. For details, see [3, 4] and [5].To state our main result, let C be an abelian category. Let X and Y be its sub-categories. Set X ∗ Y to be the subcategory consisting of objects Z such that thereis a short exact sequence 0 −→ X −→ Z −→ Y −→ X ∈ X and Y ∈ Y ,and it is called the extension subcategory of Y by X . Note that the operation “ ∗ ” onsubcategories is associative. Recall that an abelian category C has enough projectiveobjects , if for each object M there is an epimorphism P −→ M with P projective.The following result is due to Gentle and Todorov [6], which extends the corre-sponding results in artin algebras and coherent rings, obtained by Sikko and Smalø(see [11, Theorem 2.6] and [12]). Theorem 1.1. (Gentle-Todorov [6, Theorem 1.1, ii)])
Let C be an abelian categorywith enough projective objects. Assume that both X and Y are covariantly finitesubcategories in C . Then the extension subcategory X ∗ Y is covariantly finite.
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 10501041and 10601052). The author also gratefully acknowledges the support of K. C. Wong EducationFoundation, Hong Kong.E-mail: [email protected].
Proof.
The proof is also due to Gentle and Todorov, and here we just include it foran inspiration of the proof of Theorem 1.4. Note that the argument here resemblesthe one in the proof of [8, Lemma 1.3].Assume that M ∈ C is an arbitrary object. Take its left Y -approximation y M : M −→ Y M with Y M ∈ Y . By assumption the category C has enough projectiveobjects, we may take an epimorphism π M : P −→ Y M with P projective. Considerthe short exact sequence 0 −→ K −→ M ⊕ P ( y M , π M ) −→ Y M −→
0. Take a left X -approximation x K : K −→ X K of K . We form a pushout and then get the followingcommutative exact diagram0 K x K M ⊕ P ( z M , b M )( y M , π M ) Y M ∗ )0 X K i M Z M Y M . Note that Z M ∈ X ∗ Y . We claim that the morphism z M : M −→ Z M is a left X ∗ Y -approximation of M . Then we are done.To see this, assume that we are given a morphism f : M −→ Z , and Z ∈ X ∗ Y ,that is, there is an exact sequence 0 −→ X i −→ Z π −→ Y −→ X ∈ X and Y ∈ Y . Since y M is a left Y -approximation, then the composite morphism π ◦ f factorsthrough y M , say there is a morphism c : Y M −→ Y such that π ◦ f = c ◦ y M . Considerthe composite morphism P π M −→ Y M c −→ Y and the epimorphism π : Z −→ Y . Since P is projective, we have a morphism b : P −→ Z such that π ◦ b = c ◦ π M . Hence wehave the following commutative exact diagram.0 K a M ⊕ P ( f, b )( y M , π M ) Y Mc X i Z π Y . Since x K is a left X -approximation, the morphism a factors through x K , say wehave a = a ′ ◦ x K with a ′ : X K −→ X . Then by the universal mapping property ofthe pushout square ( ∗ ), there is a unique morphism h : Z M −→ Z such that h ◦ i M = i ◦ a ′ , and h ◦ ( z M , b M ) = ( f, b ) . In particular, the morphism f factors through z M , as required. (cid:4) Remark 1.2.
The same proof yields another version of Gentle-Todorov’s theorem([6, Theorem 1.1, i)]): Let C be an abelian category, X and Y covariantly finitesubcategories in C . Assume further that Y is closed under subobjects. Then theextension subcategory X ∗ Y is covariantly finite. (In the proof of this case, y M couldbe assumed to be epic, and then we just take P = 0.) (cid:4) Example 1.3.
We remark that Gentle-Todorov’s theorem may fail if the abeliancategory has not enough projective objects. To give an example, let k be a field, and N EXTENSIONS OF COVARIANTLY FINITE SUBCATEGORIES 3 let Q be the following quiver · α β · where ¯ α = { α i } i ≥ is a family of arrows indexed by positive numbers. Recall thata representation of Q , denoted by V = ( V , V ; V ¯ α , V β ), is given by the followingdata: two k -spaces V and V attached to the vertexes, and V ¯ α = ( V α i ) i ≥ , V α i : V −→ V and V β : V −→ V k -linear maps attached to the arrows. A morphismof representations, denoted by f = ( f , f ) : V −→ V ′ , consists of two linear maps f i : V i −→ V ′ i , i = 1 ,
2, which are compatible with the linear maps attached to thearrows. Denote by C the category of representations V = ( V , V ; V ¯ α , V β ) of Q suchthat dim k V = dim k V + dim k V < ∞ and V α i are zero for all but finitely many i ’s.Then C is an abelian category with finite-dimensional Hom spaces.Denote by S i the one-dimensional representation of Q seated on the vertex i withzero maps attached to all the arrows, i = 1 ,
2. Consider the two-dimensional repre-sentation M = ( M = k, M = k ; M ¯ α = 0 , M β = 1). Denote by X (resp. Y ) thesubcategory consisting of direct sums of copies of S (resp. M ). Then both X and Y are covariantly finite in C . However we claim that Z = X ∗ Y is not covariantlyfinite.In fact, the representation S does not have a left Z -approximation. Otherwise,assume that φ : S −→ V is a left Z -approximation. Take i >> V α i isa zero map. Consider the following three-dimensional representation W = ( W = (cid:18) kk (cid:19) , W = k ; W α i = (cid:18) (cid:19) , W α i = 0 for i = i , W β = (1 , . We have a non-split exact sequence of representations 0 −→ S −→ W −→ M −→ W ∈ Z . Note that Hom C ( S , W ) ≃ k . Hence there is a morphism f = ( f , f ) : V −→ W such that f ◦ φ = 0. However this is not possible. Notethat W α i ( f ( V )) = f ( V α i ( V )) = 0 by the choice of i , and that Ker W α i =Ker W β , we obtain that W β ( f ( V )) = 0. Note that both the representations S and M satisfy that the map attached to the arrow β is surjective, and then by SnakeLemma we infer that every representation in Z has this property, in particular,the representation V has this property, that is, V = V β ( V ). Hence we have 0 = W β ( f ( V )) = f ( V β ( V )) = f ( V ), and thus we deduce that f = 0. This will forcethat the composite S φ −→ V f −→ W is zero. (cid:4) We also have a triangulated version of Gentle-Todorov’s theorem. Let C be atriangulated category with the translation functor denoted by [1]. For triangulatedcategories, we refer to [13, 7]. Let X , Y be its subcategories. Set X ∗ Y to be the extension subcategory , that is, the subcategory consisting of objects Z such that thereis a triangle X −→ Z −→ Y −→ X [1] with X ∈ X and Y ∈ Y . Again this operation“ ∗ ” on subcategories is associative by the octahedral axiom (TR4). Then we havethe following result. Theorem 1.4.
Let C be a triangulated category. Assume that both X and Y are co-variantly finite subcategories in C . Then the extension subcategory X ∗Y is covariantlyfinite.
XIAO-WU CHEN
Proof.
As we noted above, the proof here is a triangulated version of the proof ofGentle-Todorov’s theorem. Assume that M ∈ C is an arbitrary object. Take its left Y -approximation y M : M −→ Y M with Y M ∈ Y . Form a triangle K k −→ M y M −→ Y M −→ K [1]. Take a left X -approximation x K : K −→ X K of K .Recall from [7, Appendix] that the octahedral axiom (TR4) is equivalent to theaxioms (TR4’) and (TR4”). Hence we have a commutative diagram of triangles K kx K M y M z M Y M K [1] x K [1] ( ∗∗ ) X K i M Z M Y M X K [1]where the square ( ∗∗ ) is a homotopy cartesian square , that is, there is a triangle K ( kxK ) −→ M ⊕ X K ( z M , − i M ) −→ Z M K [1] . (1.1)Note that Z M ∈ X ∗ Y . We claim that the morphism z M : M −→ Z M is a left X ∗ Y -approximation of M . Then we are done.To see this, assume that we are given a morphism f : M −→ Z , and Z ∈ X ∗ Y ,that is, there is a triangle X i −→ Z π −→ Y −→ X [1] with X ∈ X and Y ∈ Y . Since y M is a left Y -approximation, then the composite morphism π ◦ f factors through y M , say there is a morphism c : Y M −→ Y such that π ◦ f = c ◦ y M . Hence by theaxiom (TR3), we have a commutative diagram K ka M y M f Y Mc K [1] a [1] X i Z π Y X [1]Since x K is a left X -approximation, the morphism a factors through x K , say wehave a = a ′ ◦ x K with a ′ : X K −→ X . Hence ( i ◦ a ′ ) ◦ x K = f ◦ k , and thus( f, − i ◦ a ′ ) ◦ (cid:0) kx K (cid:1) = 0. Applying the cohomological functor Hom C ( − , Z ) to thetriangle (1.1), we deduce that there is a morphism h : Z M −→ Z such that h ◦ ( z M , − i M ) = ( f, − i ◦ a ′ ) . In particular, the morphism f factors through z M , as required. (cid:4) Applications of Gentle-Todorov’s Theorem
In this section, we apply Gentle-Todorov’s theorem to the representation theoryof artin algebras. We obtain short proofs of a classical result by Ringel and a recentresult by Krause and Solberg.Let A be an artin algebra, A -mod the category of finitely generated left A -modules.Dual to the notions of left approximations and covariantly finite subcategories, wehave the notions of right approximations and contravariantly finite subcategories . Asubcategory is called functorially finite , it is both covariantly finite and contravari-antly finite. All these properties are called homologically finiteness properties . N EXTENSIONS OF COVARIANTLY FINITE SUBCATEGORIES 5
We need more notation. Let
X ⊆ A -mod be a subcategory. Set add X to be its additive closure , that is, the subcategory consisting of direct summands of modulesin X . Note that the subcategory X has these homological finiteness properties ifand only if add X does. Let r ≥ X a subcategory of A -mod. Set F r ( X ) = X ∗ X ∗ · · · ∗ X (with r -copies of X ). Hence a module M lies in F r ( X ) if and only if M has a filtration of submodules 0 = M ⊆ M ⊆ M ⊆ · · · ⊆ M r = M with eachfactors M i /M i − in X .Note that the abelian category A -mod has enough projective and enough injectiveobjects, and thus Gentle-Todorov’s theorem and its dual (on contravariantly finitesubcategories) hold. Thus the following result is immediate. Corollary 2.1. ([11, Corollary 2.8])
Let r ≥ and X a subcategory of A -mod .Assume that X is covariantly finite (resp. contravariantly finite, functorially finite).Then the subcategories F r ( X ) and add F r ( X ) are covariantly finite (resp. contravari-antly finite, functorially finite). Recall that a subcategory X in A -mod is said to be a finite subcategory providedthat there is a finite set of modules X , X , · · · , X r in X such that each module in X is a direct summand of direct sums of copies of X i ’s. Finite subcategories arefunctorially finite ([3, Proposition 4.2]). Let r, n ≥ S = { X , X , · · · , X n } a finite set of modules, denote by S ⊕ the subcategory consisting of direct sums ofcopies of modules in S ; for each n ≥
1, set F r ( S ) = F r ( S ⊕ ). Note that S ⊕ isa finite subcategory and thus a functorially finite subcategory. The following is adirect consequence of Corollary 2.1. Corollary 2.2.
Let r, n ≥ , and S = { X , X , · · · X n } a finite set of A -modules.Then the subcategories F r ( S ) and add F r ( S ) are functorially finite. Let n ≥ S be as above. Ringel introduces in [9] the subcategory F ( S ) to bethe subcategory consisting of modules M with a filtration of submodules 0 = M ⊆ M ⊆ M ⊆ · · · ⊆ M r = M with r ≥ M i /M i − belonging to S .One observes that F ( S ) = S r ≥ F r ( S ). Then we obtain the following classical resultof Ringel with a short proof. Corollary 2.3. (Ringel, [9, Theorem 1] and [10])
Let n ≥ and S = { X , X , · · · , X n } a finite set of A -modules. Assume that Ext A ( X i , X j ) = 0 for i ≤ j . Then the subcat-egory F ( S ) is functorially finite. Proof.
First note the following factors-exchanging operation : let M be a modulewith a filtration of submodules0 = M ⊆ M ⊆ · · · ⊆ M i − ⊆ M i ⊆ M i +1 ⊆ · · · ⊆ M r = M, and we assume that Ext A ( M i +1 /M i , M i /M i − ) = 0 for some i , then M has a newfiltration of submodules0 = M ⊆ M ⊆ · · · ⊆ M i − ⊆ M ′ i ⊆ M i +1 ⊆ · · · ⊆ M r = M exchanging the factors at i , that is, M ′ i /M i − ≃ M i +1 /M i and M i +1 /M ′ i ≃ M i /M i − .We claim that F ( S ) = F n ( S ). Then by Corollary 2.2 we are done. Let M ∈ F ( S ).By iterating the factors-exachanging operations, we may assume that the module M XIAO-WU CHEN has a filtration 0 = M ⊆ M ⊆ M ⊆ · · · ⊆ M r = M such that there is a sequenceof numbers 1 ≤ r ≤ r ≤ · · · ≤ r n = r satisfying that the factors M j /M j − ≃ X i for r i − + 1 ≤ j ≤ r i (where r = 0). Because of Ext A ( X i , X i ) = 0, we deducethat M r i /M r i − +1 is a direct sum of copies of X i for each 1 ≤ i ≤ n . Therefore M ∈ F n ( S ), as required. (cid:4) We will give a short proof to a surprising result recently obtained by Krause andSolberg [8]. Note that their proof uses cotorsion pairs on the category of infinitelength modules essentially, while our proof uses only finite length modules. Recallthat a subcategory X of A -mod is resolving if it contains all projective modules and itis closed under extensions, kernels of epimorphims and direct summands ([1, p.99]). Corollary 2.4. (Krause-Solberg, [8, Corollary 0.3])
A resolving contravarianly finitesubcategory of A -mod is covariantly finite, and thus functorially finite, Proof.
Let
X ⊆ A -mod be a resolving contravariantly finite subcategory. As-sume that { S , S , · · · , S n } is the complete set of pairwise nonisomorphic simple A -modules, and take, for each i , the minimal right X -approximation X i −→ S i . Set S = { X , X , · · · , X n } . Denote by J the Jacobson idea of A , and assume that J r = 0for some r ≥
1. Thus every modules M has a filtration 0 = M ⊆ M ⊆ · · · ⊆ M r = M with semisimple factors. Hence by [2, Propostion 3.8], or more precisely by theproof [2, Proposition 3.7 and 3.8], we have that X = add F n ( S ). By Corollary 2.2the subcategory X is functorially finite. (cid:4) Let us end with an example of functorially finite subcategories.
Example 2.5.
Let A be an artin algebra and I a two-sided ideal of A such that thequotient algebra A/I is of representation finite type , that is, there are only finitelymany isoclasses of (finitely generated) indecomposable A/I -modules. For example,the Jacobson ideal J satisfies this condition. Let r ≥
1, and let X r be the subcategoryof A -mod consisting of modules annihilated by I r . We claim that the subcategory X r is functorially finite in A -mod.To see this, first note that the subcategory X could be identified with A/I -mod,and hence by assumption X is a finite subcategory, and thus functorially finite in A -mod. Then it is a pleasant exercise to check that X r = F r ( X ). By Corollary 2.1we deduce that the subcategory X r is functorially finite. Acknowledgement.
The author would like to thank Prof. Henning Krause verymuch, who half a year ago asked him to give a direct proof to their surprising result[8, Corollary 0.3]. Thanks also go to Dr. Yu Ye for pointing out the references [11, 12]and special thanks go to Prof. Apostolos Beligiannis who kindly pointed out to theauthor that Theorem 1.1 and the proof were originally due to Gentle and Todorov.
References [1]
M. Aulsnader and M. Bridger,
Stable Module Theory, Mem. Amer. Math. Soc. , Amer.Math. Soc., Providence, R. I., 1969.[2] M. Auslander and I. Reiten,
Stable equivalence of dualizing R-varieties,
Adv. Math. (1974), 306-366.[3] M. Auslander and S.O. Smalø,
Preprojective modules over artin algebras,
J. Algebra (1980), 61-122. N EXTENSIONS OF COVARIANTLY FINITE SUBCATEGORIES 7 [4]
M. Auslander and S.O. Smalø,
Almost split sequences in subcategories,
J. Algebra (1981), 426-454.[5] E.E. Enochs , Injective and flat covers, envelopes and resolvents,
Israel J. Math. (1981),189-209.[6] R. Gentle and G. Todorov,
Extensions, kernels and cokernels of homologically finite sub-categories , in: Representation theory of algebras (Cocoyoc, 1994), 227-235, CMS Conf. Proc. , Amer. Math. Soc., Providence, RI, 1996.[7] H. Krause,
Derived categories, resolutions, and Brown representability, in: Interactions be-tween homotopy theory and algebra, 101-139, Contemp. Math. , Amer. Math. Soc., Prov-idence, RI, 2007.[8]
H. Krause and Ø. Solberg , Applications of cotorsion pairs,
J. London Math. Soc. (2) (2003), 631-650.[9] C.M. Ringel , The category of mofules with good filtrations over a quasi-hereditary algebrashas almost split sequences , Math. Z. (1991), 209-223.[10]
C.M. Ringel,
On contravariantly finite subcategories , Proceedings of the Sixth Inter. Confer.on Representations of Algebras (Ottawa, ON, 1992), 5 pp., Carleton-Ottawa Math. LectureNote Ser. , Carleton Univ., Ottawa, ON, 1992.[11] S.A. Sikko and S.O. Smalø,
Extensions of homological finite subcategories,
Arch. Math. (1993), 517-526.[12] S.A. Sikko and S.O. Smalø,
Coherent rings and homologically finite subcategories,
Math.Scand. (2)(1995), 175-183.[13] J.L. Verdier,
Cat´egories d´eriv´ees, etat 0,
Springer Lecture Notes569