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2. preliminaries In this section we recall some basic results about localization of abelian categories and locally finitely presented categories.

2.1. Localization of abelian categories. In the following we recall localisation theory of abelian categories, for reader can find proof in textbooks or Gabriel thesis [3]. Let A be an . A subcategory C of A is called a Serre subcategory if for any exact sequence

0 → A1 → A2 → A3 → 0 we have that A2 ∈ C if and only if A1 ∈ C and A3 ∈ C. In this case we have the quotient A category that is by definition localisation of A with respect to the class of all morphisms C A f : X → Y such that Ker(f), Coker(f) ∈ C. is an abelian category and there exist a C universal A q : A −→ C such that q(C) = 0 for all C ∈ C, and any exact functor F : A→D annihilating C where D is abelian must factor uniquely through q. A Serre subcategory C⊆A is called a localizing subcategory if the canonical functor A A e : A→ admits a right adjoint r : → A. The right adjoint r is called the section C C functor, which always is fully faithful. Note that a localising subcategory is closed under all coproducts which exist in A. Conversely a Serre subcategory C of a A wich is closed under coproducts is a localizing subcategory, and in this case A the is also a Grothendieck category. C Let C be a Serre subcategory of an abelian category A. Recall that an object A ∈ A is called C-closed if for every morphism f : X → Y with Ker(f) ∈ C and Coker(f) ∈ C ⊥ we have that HomA(f, A) is bijective. Denote by C the full subcategory of all C-closed objects, the following result is well known. Theorem 2.1. Let C be a Serre subcategory of an abelian category A. The following statements hold: (i) We have C⊥ = {A ∈A| Hom(C, A)=0=Ext1(C, A)}. ⊥ (ii) For B ∈A and A ∈ C , the natural map qB,A : HomA(B, A) → Hom A (q(B), q(A)) C is invertible. A (iii) If C is a localizing subcategory, the restriction q : C⊥ → is an equivalence of C categories. (iv) If C is localising and A has injective envelopes, then C⊥ has injective envelopes .and the inclusion functor C⊥ ֒→A preserves injective envelopes The following definition is borrowed from [13]. HOMOLOGICALTHEORYOFSERREQUOTIENT 3

Definition 2.2. Let A be an abelian category and X be a subcategory of A. For a ⊥ ⊥ positive integer k we denote by X k the full subcategory of A defined by X k = {A ∈ ⊥ A| Ext0,...,k(X , A)=0}. k X is defined similarly [13]. Note that by Theorem 2.1 for a ⊥ ⊥ Serre subcategory C of A we have C 1 = C .

Let B be an abelian category and S be a localizing subcategory of A. Denote the localization functor and the section functor with e and r respectively, we have the following diagram of functors.

e B B S r

Proposition 2.3. Let B be a Grothendieck category, S a localizing subcategory of B, and k be a positive integer. For an object A ∈A consider the following statements. ⊥ (i) A ∈ S k+1 . (ii) There exist an injective resolution

(2.1) 0 → A → r(I0) → r(I1) →···→ r(Ik+1)

i B for A, where I ’s are injective object in S . i i (iii) The natural map eX,A : ExtB(X, A) → Ext B (e(X), e(A)) is invertible, for every S X ∈ B and 0 ≤ i ≤ k. Then (i)⇒(ii)⇒(iii). And if B has enough projective, then (iii)⇒(i).

⊥ Proof. (i)⇒(ii) Let A ∈ S k+1 , and A → I0 be the injective envelope of A in B. By applying the functor HomB(S, −) for an arbitrary object C ∈ S, to the short exact sequence

0 → A → I0 → Ω−1A → 0 we obtain the exact sequence

0 −1 0 → HomB(S, A) → HomB(S,I ) → HomB(S, Ω A) 1 1 0 1 −1 → ExtB(S, A) → ExtB(S,I ) → ExtB(S, Ω A) 2 → ExtB(S, A).

1 2 0 By assumption HomB(S, A) = ExtB(S, A) = ExtB(S, A) = 0. Because I is an essential 0 0 −1 ⊥ extension of A we have that HomB(C,I ) = 0. Thus I , Ω A ∈ S . By repeating this argument for Ω−1A and using the dimension shifting argument we obtain an injective coresolution as (2.1). (ii)⇒(iii) Apply the functor HomB(S, −) to the injective coresolution (2.1). and use the fact that (e, r) is a adjoint pair ⊥k k+1 (iii)⇒(i) Obviously A ∈ S , so it remains to show that ExtB (S, A) = 0. For an arbitrary object S ∈ S consider exact sequence 0 → B → P → S → 0 where P is a 4 RAMIN EBRAHIMI B projective object. Because P ∼= B is we have S k+1 ∼ k ExtB (S, A) = ExtB(B, A) k =∼ Ext B (B, A) S k =∼ Ext B (P, A) S k =∼ ExtB(P, A)=0. 

Let A be an abelian category. Recall that Mod A is the category of all additive con- travariant functors from A to the category of all abelian groups. It is an abelian category with all limits and colimits, which are defined point-wise. A functor F ∈ Mod A is called finitely presented(or coherent) if there exist an exact sequence Hom(−,X) → Hom(−,Y ) → F → 0 in Mod A. We denote by mod A the full subcategory of Mod A consist of finitely presented functors. It is a well known result that mod A is an abelian category because A has ,kernels. And the inclusion mod A ֒→ Mod A is an exact functor. By the Yoneda’s lemma representable functors are projective and the direct sum of all representable functors

LX∈M Hom(−,X), is a generator for Mod A. Thus Mod A is a Grothendieck category. Definition 2.4. (i) A functor F ∈ Mod A is called weakly effaceable, if for each object X ∈ A and x ∈ F (X) there exists an epimorphism f : Y → X such that F (f)(x) = 0. We denote by Eff(A) the full subcategory of all weakly effaceable functors. (ii) A functor F ∈ mod A is called effaceable, if there exist an exact sequence Hom(−,Y ) → Hom(−,X) → F → 0 such that Y → X is an epimorphism. We denote by eff(A) the full subcategory of all effaceable functors. (i) A functor F ∈ Mod A is called a left exact functor, if for each short exact sequence 0 → X → Y → Z → 0 in A the sequence of abelian groups 0 → F (Z) → F (Y ) → F (X) is exact. We denote by L(A) the subcategory of all left exact functors. Proposition 2.5. Let A be a small abelian category. (i) Eff(A) is a localizing subcategory of ModA. (ii) We have Mod A L(A)= Eff(A)⊥ ≃ . Eff(A)

Proof. This is an standard result, see for example [5].  We denote by i : A→L(A) the composition of the Yoneda functor A → ModA and ModA ⊥ the localization functor ModA→ Eff(A) ≃ Eff(A) = L(A). HOMOLOGICALTHEORYOFSERREQUOTIENT 5

2.2. Locally coherent categories. In this subsection we recall some of the basic prop- erties of Locally finitely presented categories, for details and more information the reader is referred to [1]. Recall that a non-empty category I is said to be filtered provided that for each pair of objects λ1,λ2 ∈ I there are morphisms ϕi : λi → µ for some µ ∈ I, and for each pair of morphisms ϕ1,ϕ2 : λ → µ there is a morphism ψ : µ → ν with ψϕ1 = ψϕ2. Let A be an additive category, I a small filtered category and X : I→A be an additive functor. As usual we use the term direct for colimit of X when I is filtered. And ∈IX we denote it by−−→ Limi i. Definition 2.6. Let B be an additive category with direct limits . (i) An object X ∈ B is called finitely presented (finitely generated) provided that for ∈I X every direct limit Lim−−→i i in B the natural morphism B X,Y B X, Y −−→Lim Hom ( i) → Hom ( −−→Lim i) is an isomorphism (a monomorphism). The full subcategory of finitely presented objects of A is denoted by fp(A). (ii) B is called a locally finitely presented category if fp(B) is skeletally small and every object in B is a direct limit of objects in fp(B). (iii) Let B be a locally finitely presented abelian category. B is said to be locally coherent provided that finitely generated subobjects of finitely presented objects are finitely presented. We want to recall a concrete description of locally finitely presented additive categories, which is due to Crawley-Boevey. Let C be a small additive category, recall that there is a tensor product bifunctor op Mod C⊗C Mod C −→ Ab.

(F,G) 7−→ F ⊗C G

A functor F ∈ Mod C is said to be flat provided that F ⊗C − is an exact functor. We denote by Flat(C) the full subcategory of Mod C consist of flat functors. A theorem of Lazard state that an R-module is flat if and only if it is a direct limit of finitely generated free modules. Lazard theorem has been generalized to functors by Oberst and Rohrl [9]. Indeed a functor F ∈ Mod C is flat if and only if F is a direct limit of representable functors. Lemma 2.7. Let A be an abelian category and F ∈ Mod A. The following statements are equivalence. (a) F is a flat functor. (b) F is a left exact functor. In other words Flat(A)= L(A). Proof. See [5].  Theorem 2.8. ([1, Theorem 1.4]) (a) If C is a skeletally small additive category, then Flat(C) is a locally finitely pre- sented category, and fp(Flat(C)) consists of the direct summands of representable 6 RAMIN EBRAHIMI

functors. if C has split idempotents then the Yoneda functor h : C → fp(Flat(C)) is an equivalence. (b) If B is a locally finitely presented category then fp(B) is a skeletally small additive category with split idempotents, and the functor g : B −→ Flat(fp(B))

M 7−→ Hom(−, M)|fp(B) is an equivalence. Proposition 2.9. (i) Every locally finitely presented abelian category is a Grothendieck category. (ii) A locally finitely presented abelian category B is locally coherent if and only if fp(B) is an abelian category. Proof. We refer the reader to [1] for the statement (i) and to [14] for (ii).  In the proof of the main theorem we need the following proposition. Proposition 2.10. Let A be an abelian category with enough projective. Then L(A) is a locally coherent category with enough projective. Proof. By Proposition 2.9 and Lemma 2.7 L(A) is a locally coherent category. so it remains to show that it has enough projective. First note that if P be a projective object in A then HP is a projective object in L(A). Indeed for each epimorphism F → HP in L(A), by Proposition 2.5 it’s cokernel is weakly effaceable. Thus there exists an epimorphism X → P and the commutative diagram

HX HP

F HP

Because HX → HP is a retraction, F → HP is also a retraction. Now for an arbitrary ob- ։ ject F ∈L(A), there is an epimorphism Li∈I HXi F , and if we choose an epimorphism Pi ։ Xi in A for each i ∈ I the result follows. 

3. Proof of the main theorem Let A be a small abelian category and C be a Serre subcategory of A. By Theorem 2.8 Mod A L(A) ≃ is a locally coherent category and the essential image of the canoonical Eff(A) functor i : A −→L(A) denoted by i(C) is the subcategory of finitely presented objects. Thus by [7, Theorem 2.8] −−→ i(C), the subcategory of L(A) consist of all direct limits of objects in i(C) is a localizing L(A) subcategory of L(A). Also by [6, Proposition A5] −−→ is a locally coherent category i(C) HOMOLOGICALTHEORYOFSERREQUOTIENT 7 A and it’s subcategory of finitely presented objects is equivalent to . Thus we have the C diagram of functors

q A A C

i j

e L(A) A L(A) −−→ ≃L( ) i(C) C r (3.1) satisfies (i) q and e are localization functors. (ii) (e, r) is an adjoint pair and r is fully faithful. (iii) Both of vertical functors i and j are the canonical functor from an abelian category to the abelian category of left exact functors.

Lemma 3.1. Let A be a small abelian category. The canonical functor i : A −→L(A) is an Ext-preserving functor. i.e for every two object A, B ∈A and every positive integer i i i the natural map ExtA(A, B) → ExtL(A)(i(A), i(B)) is invertible.

Proof. See the section Application of [10] on page 367. 

The following proposition is a generalization of [7, Corollary 2.11] for arbitrary positive integer k. Krause prove this result for k = 1.

Proposition 3.2. Let B be a locally coherent category with enough projective, C be a Serre subcategory of A = fp(B) and k be a positive integer. Then as of B ⊥ −→ ⊥ we have C k =( C ) k .

⊥ −→ ⊥ −→ ⊥ ⊥ Proof. By [7, Corollary 2.11] we have C 1 = ( C ) 1 . Obviously ( C ) k ⊆ C k . For the ⊥ −→ ⊥ converse let X ∈ C k . We most show that X ∈ ( C ) k . To this end by Proposition 3.2 it is enough to show that there is an injective resolution

0 → X → I0 → I1 →···→ Ik

−→ ⊥ ⊥ such that ∀i we have Ii ∈ ( C ) 1 = C 1 . Let X → I0 be the injective envelope of X in B. By applying the functor HomB(C, −) for an arbitrary object C ∈ C, to the short exact sequence

0 → X → I0 → Ω−1X → 0 8 RAMIN EBRAHIMI we obtain the exact sequence 0 −1 0 → HomB(C,X) → HomB(C,I ) → HomB(C, Ω X) 1 1 0 1 −1 → ExtB(C,X) → ExtB(C,I ) → ExtB(C, Ω X) 2 → ExtB(C,X). 1 2 0 By assumption HomB(C,X) = ExtB(C,X) = ExtB(C,X) = 0. Because I is an essential 0 0 −1 ⊥ −→ ⊥ extension of X we have that HomB(C,I ) = 0. Thus I , Ω X ∈ C = ( C ) . By repeating this argument for Ω−1X and using the dimension shifting argument, similar to the proof of Proposition we obtain the desire injective coresolution.  Now we are ready to prove Theorem 1.1. ⊥ Proof of Theorem 1.1:Let A ∈ C k+1 and X be an arbitrary object in A. Having the Diagram (3.1) in mind we see that

iX,A i ∼ i ExtA(X, A) = ExtL(A)(i(X), i(A)) eX,A ∼ i = Ext L(A) (ei(X), ei(A)) −→ C because jX,AoqX,A = eX,AoiX,A we have that jX,AoqX,A is invertible, and because by Lemma 3.1 jX,A is invertible, qX,A is also invertible. Conversely assume that qX,A is invertible for every X ∈A and i ∈{0, 1, ··· ,k}. Bya similar argument like the proof (iii)⇒(i) of Proposition 3.2 the result follows. We state the dual of Theorem 1.1. Theorem 3.3. Let A be an abelian category with enough injective, C a Serre subcategory A of A, k be a positive integer, and q : A→ C be the quotient functor. For an object A ∈A the following conditions are equivalent. ⊥ (i) A ∈ k+1 C. i i (ii) The natural map qA,X : ExtA(A, X) → Ext A (q(A), q(X)) is invertible, for every C X ∈A and 0 ≤ i ≤ k. Proof. Apply Theorem 1.1 to the opposite category Aop. 

4. Applications Let n be a positive integer and M be a small n-abelian category in the sense of [4]. Denote by eff(M) the full subcategory of mod M consist of effaceable functors, i.e the functors F ∈ mod M with a projective presentation (−,f) (4.1) (−,Y n) −→ (−,Y n+1) −→ F → 0 for some epimorphism f : Y n → Y n+1. In [2, 8] it has been shown that eff(M) is a Serre subcategory of mod M, the composition of functors mod M M −→ mod M −→ eff(M) X 7−→ (−,X) 7−→ (−,X) HOMOLOGICALTHEORYOFSERREQUOTIENT 9 mod M is fully faithful and it’s essential image is an n-cluster tilting subcategory of . eff(M) The hardest step is maybe proving that the essential image is n-rigid, i.e for every k ∈ k {1, ··· , n − 1} and X,Y ∈ M we have Ext mod M (HX ,HY ) = 0. In the sequel we give a eff(M) different proof using Theorem 1.1. Let F ∈ eff(M) and consider a projective presentation (4.1). Because f is an epimor- phism, by the axioms of n-abelian categories f fits into an n-exact sequence f Y 0 → Y 1 →···→ Y n → Y n+1. By the definition of n-exact sequences we have the exact sequence

0 → HY 0 → HY 1 →···→ HY n → HY n+1 → F → 0. which is a projective resolution of F . Now let X ∈ M. Applying Hommod-M(−,HX ) to the projective resolution of F , we get the complex

0 → Hom(HY n+1 ,HX ) → Hom(HY n ,HX ) →···

→ Hom(HY 1 ,HX ) → Hom(HY 0 ,HX ) → 0 which is by Yoneda’s lemma isomorphic to the complex 0 → Hom(Y n+1,X) → Hom(Y n,X) →···→ Hom(Y 1,X) → Hom(Y 0,X) that by the definition of n-exact sequences is an exact sequence of abelian groups. Thus k we see that for every k ∈{0, 1, ··· , n} we have Ext mod M (F,HX) = 0. Because F was an eff(M) ⊥n arbitrary effaceable functor, for every X ∈ M, HX ∈ eff(M) . Therefore by Theorem 1.1 for every k ∈{1, ··· , n − 1} and X,Y ∈M we have k ∼ k Ext mod M (HX ,HY ) = Extmod M(HX ,HY ) = 0. eff(M) acknowledgements This research was in part supported by a grant from IPM (No. 1400180047).

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[13] C. Psaroudakis, Homological theory of recollements of abelian categories, J. Algebra, 398 (2014), 63–110. [14] J.-E. Roos, Locally noetherian categories, In: category Theory, Homology Theory and their Appli- cations II, Springer Lecture Notes in Math. 92 (1969) 197–277.

Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, P.O. Box: 81746-73441, Isfahan, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran Email address: [email protected] / [email protected]