Closure properties associated to natural equivalences

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Indagationes Mathematicae 24 (2013) 403–414 www.elsevier.com/locate/indag

Closure properties associated to natural equivalences Flaviu Pop ∗ “Babes¸-Bolyai” University, Faculty of Economics and Business Administration, str. T. Mihali, nr. 58-60, 400591, Cluj-Napoca, Romania Received 25 October 2011; accepted 20 December 2012 Communicated by M. Crainic

Abstract Given a pair of adjoint functors F : A  B : G, we study some closure properties of some full subcategories A and B such that the restrictions F : A  B : G induce an equivalence. c 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝ Keywords: Adjoint functors; Static object; Natural equivalence; Grothendieck category

1. Introduction The study of equivalences induced by pairs of adjoint covariant functors is an important topic. Using Hom and ⊗ functors, this topic developed important concepts in the Module Theory. These concepts, as tilting and star module, are used in the study of representable equivalences (see [12,26] for complete surveys on the subjects). Moreover, this kind of study is also useful in a more general setting, in order to apply the results to other kind of categories. For instance, Casta˜no-Iglesias, G´omez-Torrecillas and Wisbauer applied the study of adjoint pairs of functors between Grothendieck categories to special categories of graded modules or comodules [11]. Marcus and Modoi, in [20], also used other kind of equivalences in order to study categories of graded modules. Colpi [13], Gregorio [18] and Rump [24] constructed a general theory of tilting objects in various kind of categories. Recently, Bazzoni [5] considers some particular categories of fractions (which, in general, have no infinite direct sums) in order to describe the

∗ Tel.: +40 740166153.

E-mail addresses: [email protected], [email protected]. c 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights 0019-3577/$ - see front matter ⃝ reserved. doi:10.1016/j.indag.2012.12.006

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F. Pop / Indagationes Mathematicae 24 (2013) 403–414

classes involved in a tilting theorem [5, Theorem 4.5], while Breaz [8,6] studied functors and equivalences between similar categories of fractions in order to apply these results to the category of abelian groups and quasi-homomorphisms (see [1] for such an application). In the study of equivalences induced by a pair of adjoint functors, some closure properties of the classes involved in these equivalences are very important. For instance, some closure properties with respect to submodules or to quotients are used to characterize the representability of these functors [15,21] and the pairs of functors which generalize tilting modules and star modules can be characterized by some closure properties [14,24]. Furthermore, closure properties of the classes of static modules, respectively adstatic modules (i.e. the maximal classes such that the maps of adjunction are isomorphisms), with respect to submodules or to submodules of finite index are very useful in the study of abelian groups which have some flatness properties as modules over their endomorphism ring [2,7] or of the important class of S-groups [4] and their generalizations [3]. In this paper we study some closure properties which are inspired from the work of Mantese and Tonolo (see [19]) and Fuller (see [16]) on dualities. We start with a preliminary section where we present some basic notions and results. Then, in Section 3, which is the main section of the paper, we consider a pair of adjoint additive covariant functors F : A  B : G between two abelian categories. We are interested about the closure properties of some full subcategories A and B such that the restrictions F : A  B : G are equivalences. More precisely, we characterize situations when B is closed with respect to faithful factors by a closure property of A and an exactness property of F in Theorem 3.3 and Proposition 3.7. Then, in Proposition 3.8, we identify when the converse of the exactness property of F is valid. These results are applied in Section 4 for closure properties of some classes constructed starting with the class add(V ), where V = F(U ) and U is a static object. We recall that, for an object X, add(X ) denotes the class of all direct summands of finite direct sums of copies of X . If R is an unital associative ring, we denote by Mod-R (respectively, by R-Mod) the category of all right (respectively, left) R-modules. As in the papers [10,11,23], the results obtained here can be applied to the pairs of adjoint additive and covariant functors presented in Section 2. 2. Preliminaries Throughout this paper, we consider a pair of additive and covariant functors F : A  B : G between abelian categories such that G is a left adjoint to F, i.e. there are natural isomorphisms ϕ X,M : HomA (G(X ), M) → HomB (X, F(M)), for all M ∈ A and for all X ∈ B. Then, they induce two natural transformations φ : GF → 1A ,

−1 defined by φ M = ϕF(M),M (1F(M) )

θ : 1B → FG,

defined by θ X = ϕ X,G(X ) (1G(X ) ).

and

We note that F is left exact and G is right exact. Moreover, they satisfy the identities F(φ M ) ◦ θF(M) = 1F(M) and φG(X ) ◦ G(θ X ) = 1G(X ) , for all M ∈ A and for all X ∈ B. We also assume that all considered subcategories are isomorphically closed. The classical example of such a pair of functors is the following (see [25], Chapter 9):

F. Pop / Indagationes Mathematicae 24 (2013) 403–414

405

Example 2.1. Let R and S be two unital associative rings and let S Q R be an (S, R)-bimodule. Then F(−) = Hom R (Q, −) : Mod-R  Mod-S : − ⊗ S Q = G(−) and F(−) = Hom S (Q, −) : S-Mod  R-Mod : Q ⊗ R − = G(−) are pairs of additive and covariant functors. Moreover, the tensor functor − ⊗ S Q (respectively, Q ⊗ R −) is an adjoint on the left of the covariant Hom functor Hom R (Q, −) (respectively, Hom S (Q, −)).  The next two examples were presented by Casta˜no-Iglesias, G´omez-Torrecillas and Wisbauer in [11]:  Example 2.2. Let G be a group. If R = x∈G R x is a G-graded ring, we will denote by R-Modgr the category of all G-graded unital left R-modules. If M, N ∈ R-Modgr , we consider the G-graded abelian group HOM R (M, N ) whose homogeneous component at x is the subgroup of Hom R (M, N ) consisting of all R-homomorphisms f : M → N such that f (M y ) ⊆ N yx , for all y ∈ G. We note that S = HOM R (M, M) = END R (M) is a G-graded ring and M has a G-graded (R, S)-bimodule structure, in sense that Rx · M y · Sz ⊆ Mx yz , for every x, y, z ∈ G. If T is a G-graded unital left S-module, then the unital left R-module M ⊗ S T has a G-graded left R-module structure, where the homogeneous component at x is (M ⊗ S T )x = { yz=x m y ⊗ tz | m y ∈ M y , tz ∈ Tz }. For x ∈ G, we denote by M x the left R-module M endowed with a new grading given by (M x ) y = M yx , for all y ∈ G. If Q ∈ R-Modgr with END R (Q) = S, then F(−) = HOM R (Q, −) : R-Modgr  S-Modgr : Q ⊗ S − = G(−) is a pair of additive and covariant functors. Moreover, the functor Q ⊗ S − is a left adjoint to the functor HOM R (Q, −).  A more detailed approach of the example presented above is to be found in [22]. Example 2.3. Let C be a coalgebra over a commutative ring R with identity. We denote by M C the category of all right C-comodules. This category is a Grothendieck category if and only if C is flat as R-module. A right C-comodule M is called quasi-finite if the functor − ⊗ R M : Mod-R → M C has a left adjoint. If Q is a quasi-finite right C-comodule and D = h(Q, Q) is the coendomorphism coalgebra, then F(−) = −

D

Q : M D  M C : HC (Q, −) = G(−)

is a pair of additive and covariant functors, where − D Q is the cotensor functor and HC (Q, −) is the cohom functor induced by Q. Moreover, HC (Q, −) is left adjoint to − D Q.  The following example is used by Breaz in [8]. Example 2.4. Let R be a ring and Σ be a multiplicatively closed set of non-zero integers. We consider the category of fractions Z[Σ −1 ]Mod-R which has as objects all the right R-modules and if M, N ∈ Mod-R, then HomZ[Σ −1 ]Mod-R (M, N ) = Z[Σ −1 ] ⊗Z Hom R (M, N ). There is a canonical functor q : Mod-R → Z[Σ −1 ]Mod-R. By [17], every pair of adjoint functors

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F : Mod-R  Mod-S : G induces a canonical pair of adjoint functors q F : Z[Σ −1 ]Mod-R  Z[Σ −1 ]Mod-S : qG such that qF = (q F)q and qG = (qG)q (here q denotes both the canonical functors Mod-R → Z[Σ −1 ]Mod-R and Mod-S → Z[Σ −1 ]Mod-S). Starting with the setting presented in Example 2.1, we have that F(−) = qHom R (Q, −) : Z[Σ −1 ]Mod-R  Z[Σ −1 ]Mod-S : q(− ⊗ S Q) = G(−) is a pair of adjoint covariant functors.



An object M ∈ A (respectively, X ∈ B) is called φ-faithful (respectively, θ -faithful) if φ M (respectively, θ X ) is a monomorphism. We denote by Faithφ (respectively, Faithθ ) the class of all φ-faithful (respectively, θ -faithful) objects. An object M ∈ A (respectively, X ∈ B) is called φ-generated (respectively, θ -generated) if φ M (respectively, θ X ) is an epimorphism. We denote by Genφ (respectively, Genθ ) the class of all φ-generated (respectively, θ -generated) objects. An object M ∈ A (respectively, X ∈ B) is called F-static (respectively, F-adstatic) if φ M (respectively, θ X ) is an isomorphism. We denote by StatF (respectively, by AdstatF ) the class of all F-static (respectively, F-adstatic) objects. Let A be an object in A. We say that M is finitely-A-generated if there is an epimorphism An −→ M → 0, for some positive integer n. We denote by gen(A) the class of all finitelyA-generated objects. We say that M is finitely-A-cogenerated if there is a monomorphism 0 → M −→ An , for some positive integer n. We denote by cog(A) the class of all finitelyA-cogenerated objects. Lemma 2.5. The following assertions hold: (a) (b) (c) (d) (e)

F(A) ⊆ Faithθ and G(B) ⊆ Genφ ; F(StatF ) = AdstatF and G(AdstatF ) = StatF ; The class Genφ is closed with respect to factors; The class Faithθ is closed with respect to subobjects; StatF and AdstatF are closed with respect to finite direct sums and direct summands.

Moreover, if U is an F-static object with F(U ) = V then: (f) add(U ) ⊆ StatF and add(V ) ⊆ AdstatF ; (g) F(add(U )) = add(V ) and G(add(V )) = add(U ). Proof. The proofs of these assertions can be done by inspection (see [9]). For reader’s convenience we prove here (a) and (c). (a) If M ∈ A it follows, from the identity F(φ M ) ◦ θF(M) = 1F(M) , that θF(M) is a monomorphism, i.e. F(M) ∈ Faithθ . If X ∈ B then, from the identity φG(X ) ◦ G(θ X ) = 1G(X ) , it follows that φG(X ) is an epimorphism, i.e. G(X ) ∈ Genφ . (c) Let M ∈ Genφ and let N be a factor of M. Then, there is an epimorphism g : M → N , hence the following diagram GF(g)

GF(M) −−−−→ GF(N )    φ φM   N g

M −−−−→ N is commutative, i.e. g ◦ φ M = φ N ◦ GF(g). Since g and φ M are epimorphisms, it follows that φ N is an epimorphism, and thus N ∈ Genφ . 

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F. Pop / Indagationes Mathematicae 24 (2013) 403–414 f

g

Corollary 2.6. If 0 → X −→ Y −→ Z → 0 is an exact sequence in B, then ImG( f ) = KerG(g) ∈ Genφ . Proof. It is enough to observe that G is right exact and that G(X ) ∈ Genφ . f



g

Lemma 2.7. Let 0 → K −→ M −→ N → 0 be an exact sequence in A and let β be the unique morphism such that the following diagram GF( f )

G(π)

f

g

GF(K ) −−−−→ GF(M) −−−−→ G(ImF(g)) −−−−→ 0     φ β φK   M  0 −−−−→ K −−−−→ M −−−−→ N −−−−→ 0 is commutative (with exact rows). Then β = φ N ◦ G(σ ), where F(g) = σ ◦ π is the canonical decomposition of F(g). f

g

Lemma 2.8. Let 0 → K −→ M −→ N → 0 be an exact sequence in A, with M ∈ StatF and F(g) an epimorphism. Then K ∈ Genφ if and only if N ∈ StatF . F( f )

F(g)

Proof. Since F(g) is an epimorphism, the sequence 0 → F(K ) −→ F(M) −→ F(N ) → 0 is GF( f )

GF(g)

exact, hence the sequence GF(K ) −→ GF(M) −→ GF(N ) → 0 is also exact. It follows that the following diagram GF( f )

GF(g)

f

g

GF(K ) −−−−→ GF(M) −−−−→ GF(N ) −−−−→ 0     φ φ φK   M  N 0 −−−−→ K −−−−→ M −−−−→ N −−−−→ 0 is commutative with exact rows. Since M is F-static, we obtain, by Snake Lemma, that Ker(φ N ) ∼ = Coker(φ K ) and φ N is an epimorphism. Therefore K ∈ Genφ if and only if N ∈ StatF .  f

g

Lemma 2.9. Let 0 → K −→ M −→ N → 0 be an exact sequence in A, with M ∈ StatF and K ∈ Genφ . Then F(g) is an epimorphism if and only if ImF(g) ∈ AdstatF . Proof. Suppose that F(g) is an epimorphism. It follows that ImF(g) = F(N ). By Lemma 2.8, we have that N is F-static, hence F(N ) is F-adstatic. Therefore ImF(g) ∈ AdstatF . Conversely, assume that ImF(g) ∈ AdstatF . Let F(g) = σ ◦π be the canonical decomposition. By Lemma 2.7, the following diagram is commutative with exact rows GF( f )

G(π)

f

g

GF(K ) −−−−→ GF(M) −−−−→ G(ImF(g)) −−−−→ 0     φ β φK   M  0 −−−−→ K −−−−→ M −−−−→ N −−−−→ 0 where β = φ N ◦ G(σ ). Since M ∈ StatF and K ∈ Genφ , it follows, by the Snake Lemma, that β is an isomorphism, hence F(β) is also an isomorphism. We have the identities σ = 1F(N ) ◦ σ = F(φ N ) ◦ θF(N ) ◦ σ = F(φ N ) ◦ FG(σ ) ◦ θImF(g) = F(φ N ◦ G(σ )) ◦ θImF(g) = F(β) ◦ θImF(g) .

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Since F(β) and θImF(g) are isomorphisms, it follows that σ is an isomorphism, hence ImF(g) ∼ = F(N ). Therefore F(g) is an epimorphism.  3. Closure properties with respect to θ-faithful factors Proposition 3.1. Let Y be an F-adstatic object. The following statements are equivalent: (a) If Z is a θ -faithful factor of Y , then Z ∈ AdstatF ; f

g

(b) If 0 → K −→ G(Y ) −→ N → 0 is an exact sequence in A with K ∈ Genφ , then F(g) is an epimorphism. f

g

Proof. (a)⇒(b) Let 0 → K −→ G(Y ) −→ N → 0 be an exact sequence in A with K ∈ F( f )

Genφ . Then 0 → F(K ) −→ FG(Y ) −→ ImF(g) → 0 is an exact sequence, hence FG(Y )/F(K ) ∼ = ImF(g). By Lemma 2.5, ImF(g) is θ -faithful. Since Y is F-adstatic, hence ImF(g) ∼ = Y /F(K ), it follows from (a), that ImF(g) ∈ AdstatF . Applying Lemma 2.9 to the considered sequence, we obtain that F(g) is an epimorphism. (b)⇒(a) Let Z be a θ -faithful factor of Y . Then, there exists in B an exact sequence 0 → f

G(g)

g

X −→ Y −→ Z → 0. From the fact that the sequence 0 → ImG( f ) −→ G(Y ) −→ G(Z ) → 0 is exact in A with ImG( f ) ∈ Genφ , we have, according to (b), that FG(g) is an epimorphism. In the following commutative diagram Y   θY 

g

−−−−→

Z  θ Z

FG(g)

FG(Y ) −−−−→ FG(Z ) θY is an isomorphism and FG(g) is an epimorphism. It follows that θ Z is an epimorphism and, since θ Z is also a monomorphism, we have that θ Z is an isomorphism.  Corollary 3.2. Let Y be an F-adstatic object which satisfies the equivalent conditions from the previous result. If K ∈ Genφ is a subobject of G(Y ) then G(Y )/K is F-static. Proof. Let K ∈ Genφ be a subobject of G(Y ). Then we could consider the exact sequence p

i

0 → K −→ G(Y ) −→ G(Y )/K → 0 in A. From the condition (b) of Proposition 3.1, it follows that F( p) is an epimorphism. The conclusion follows by Lemma 2.8.  Theorem 3.3. Let B0 be a full additive subcategory of B consisting in F-adstatic objects and let A0 = G(B0 ). Let B be the class of all θ -faithful factors of objects in B0 and let A = {M/K | M ∈ A0 , K ∈ Genφ }. Then the following statements are equivalent: (a) F : A  B : G is an equivalence and B is closed under θ -faithful factors; (b) B ⊆ AdstatF ; f

g

(c) If 0 → K −→ M −→ N → 0 is an exact sequence in A, with M ∈ A0 and K ∈ Genφ , then F(g) is an epimorphism. f

g

Proof. (b)⇒(c) Let 0 → K −→ M −→ N → 0 be an exact sequence in A, with M ∈ A0 and K ∈ Genφ . Since M ∈ A0 , there is Y ∈ B0 such that M = G(Y ). Since Y ∈ B0 , we have that Y is F-adstatic. According to (b), we observe that Y satisfies the condition (a) of Proposition 3.1.

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F. Pop / Indagationes Mathematicae 24 (2013) 403–414 f

g

Since 0 → K −→ G(Y ) −→ N → 0 is exact with K ∈ Genφ , it follows, by condition (b) of Proposition 3.1, that F(g) is an epimorphism. (c)⇒(b) Let Z ∈ B. Then Z = Y / X is θ -faithful, for some Y ∈ B0 . Since Y ∈ B0 , we have that Y is an F-adstatic object. Thanks to (c), Y satisfies the condition (b) of Proposition 3.1. By condition (a) of Proposition 3.1, we obtain that Z ∈ AdstatF . Therefore B ⊆ AdstatF . (a)⇒(b) Since F : A  B : G is an equivalence, we have B ⊆ AdstatF . (b)⇒(a) Assume that (b) holds. Since (b) is equivalent to (c), it follows that (c) holds. From (b), we have that B ⊆ AdstatF . Next, we show that G(B) ⊆ A. Let Z ∈ B. Then f

Z is a θ -faithful factor of Y , for some Y ∈ B0 . Thus, there is an exact sequence 0 → X −→ σ

g

G(g)

Y −→ Z → 0, hence the sequence 0 → ImG( f ) −→ G(Y ) −→ G(Z ) → 0 is also exact, where σ comes from the canonical decomposition of G( f ), namely G( f ) = σ ◦ π . It follows that G(Z ) ∼ = G(Y )/ImG( f ). By Corollary 2.6, ImG( f ) ∈ Genφ , and since G(Y ) ∈ A0 , we have G(Z ) ∈ A. Therefore G(B) ⊆ A. In the following, we show that A ⊆ StatF and F(A) ⊆ B. Let N ∈ A. Then N = M/K , for f

g

some M ∈ A0 and K ∈ Genφ . Then, there exists an exact sequence 0 → K −→ M −→ N → 0 and, applying (c) to this sequence, we obtain that F(g) is an epimorphism. According to Lemma 2.8, we have N ∈ StatF . Therefore A ⊆ StatF . Since M ∈ A0 , there is Y ∈ B0 such that M = G(Y ). Given that F(M) = FG(Y ) and Y is F-adstatic, we observe that F(M) ∈ B0 . By Lemma 2.5, F(N ) is θ -faithful. Since F(N ) ∼ = F(M)/F(K ), it follows that F(N ) ∈ B. Thus F(A) ⊆ B. Summarizing the above, we conclude that F : A  B : G is an equivalence. It remains to prove that B is closed under θ -faithful factors. Let Y ∈ B and let Z be a θ -faithful factor of Y . Then, there is a subobject X of Y such that Z = Y / X . Hence we have the exact sequence i

p

0 → X −→ Y −→ Z → 0. Since Y ∈ B, there are T ∈ B0 and S a subobject of T such that j

q

Y = T /S is θ -faithful, hence we have the exact sequence 0 → S −→ T −→ Y → 0. Since p and q are epimorphisms, we have that pq : T −→ Z is an epimorphism, hence the sequence pq 0 → Ker( pq) −→ T −→ Z → 0 is exact and then Z ∼ = T /Ker( pq). Using this argument and taking into account that T ∈ B0 and Z ∈ Faithθ , we have Z ∈ B. It follows that the class B is closed under θ -faithful factors.  Example 3.4. Let U be an F-static object with F(U ) = V . Since V is F-adstatic, hence V k is F-adstatic, for all positive integers k, we could consider B0 = {V k | k ∈ N∗ }. Then A0 = {U k | k ∈ N∗ }. We observe that B = gen(V ) ∩ Faithθ and the class A consists in all objects N ∈ A such that N = U n /K with K ∈ Genφ and n ∈ N∗ . Example 3.5. Let R be an unital associative ring and let Q be a right R-module. If S = End R (Q) is the endomorphism ring of Q, then Q has a structure of (S, R)-bimodule and, as we see in Example 2.1, we have the pair F(−) = Hom R (Q, −) : Mod-R  Mod-S : − ⊗ S Q = G(−) of additive and covariant functors. Moreover, the right R-module Q R is Hom R (Q, −)-static and the right S-module SS is Hom R (Q, −)-adstatic. (i) If we set B0 = {S k | k ∈ N∗ } then we have A0 = {Q k | k ∈ N∗ }, B = gen(S) ∩ Faithθ and A consists in the class of all right R-modules N such that N = Q n /K , for some K ∈ Genφ and n ∈ N∗ ; (ii) If we set B0 = {S} then we have A0 = {Q}, B = {Z ∈ B | Z = S/ X } ∩ Faithθ and A consists in the class of all right R-modules N such that N = Q/K , for some K ∈ Genφ .

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 Example 3.6. Let G be a group and let R = x∈G Rx be a G-graded ring. Let Q ∈ R-Modgr with S = END R (Q). Then S is a G-graded ring and Q is a G-graded (R, S)-bimodule. As we saw in Example 2.2, we have the pair F(−) = HOM R (Q, −) : R-Modgr  S-Modgr : Q ⊗ S − = G(−) of additive and covariant If Q is gr-self-small, i.e. HOM   functors. R (Q, −) x , then x is F-static. Moreover, HOM (Q, x preserves coproducts of Q Q R x∈G x∈G Q )   x∈G x  x x = x∈G S . Denoting x∈G Q by U and x∈G S by V , we have: (i) If we set B0 = {V k | k ∈ N∗ } then we have A0 = {U k | k ∈ N∗ }, B = gen(V ) ∩ Faithθ and A consists in the class of all G-graded unital left R-modules N such that N = U n /K , for some K ∈ Genφ and n ∈ N∗ ; (ii) If we set B0 = {V } then we have A0 = {U }, B = {Z ∈ B | Z = V / X } ∩ Faithθ and A consists in the class of all G-graded unital left R-modules N such that N = U/K , for some K ∈ Genφ . Proposition 3.7. Let F : A  B : G be an equivalence between the full additive subcategories A and B of A and B, respectively. The following statements are equivalent: (a) B is closed under θ -faithful factors; (b) (1) A is closed with respect to factors modulo φ-generated subobjects; f

g

(2) F is exact with respect to the short exact sequences 0 → K −→ M −→ N → 0 with M ∈ A and K ∈ Genφ . Proof. Since F : A  B : G is an equivalence, we have F(A) ⊆ B ⊆ AdstatF and G(B) ⊆ A ⊆ StatF . (a)⇒(b) (1) Let M ∈ A. Then F(M) is an F-adstatic object which satisfies the condition (a) of Proposition 3.1, because B is closed under θ -faithful factors. f

g

Let K ∈ Genφ be a subobject of M. Then there is an exact sequence 0 → K −→ M −→ N → 0. Since G(F(M)) ∼ = M we obtain, by condition (b) of Proposition 3.1, that F(g) is an F( f )

F(g)

epimorphism. Hence the sequence 0 → F(K ) −→ F(M) −→ F(N ) → 0 is exact and thus F(N ) ∼ = F(M)/F(K ). Since F(N ) is θ -faithful, F(M) ∈ B and B is closed under θ -faithful factors, it follows that F(N ) ∈ B, hence GF(N ) ∈ A. On the other hand, by Lemma 2.8, N is F-static, whence GF(N ) ∼ = N . Therefore N ∈ A and this shows us that (1) is true. f

g

(2) Let 0 → K −→ M −→ N → 0 be an exact sequence, with M ∈ A and K ∈ Genφ . As F( f )

F(g)

we saw before, F(g) is an epimorphism, hence the induced sequence 0 → F(K ) −→ F(M) −→ F(N ) → 0 is exact. (b)⇒(a) Let Y ∈ B and let Z be a θ -faithful factor of Y . Then there is an exact sequence f

G(g)

g

0 → X −→ Y −→ Z → 0 in B. If follows that the sequence 0 → KerG(g) −→ G(Y ) −→ G(Z ) → 0 is also exact, hence G(Z ) ∼ = G(Y )/KerG(g). By Corollary 2.6, KerG(g) ∈ Genφ . Since G(Y ) ∈ A and since A is closed under factors modulo φ-generated subobjects, it follows that G(Z ) ∈ A, hence FG(Z ) ∈ B. On the other hand, according to (b)(2), FG(g) is an epimorphism. The following diagram Y   θY 

g

−−−−→

FG(g)

Z  θ Z

FG(Y ) −−−−→ FG(Z )

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F. Pop / Indagationes Mathematicae 24 (2013) 403–414

is commutative with θY being an isomorphism and FG(g) being an epimorphism. It follows that θ Z is an epimorphism and taking into account that Z is θ -faithful, hence θ Z is a monomorphism, we obtain that θ Z is an isomorphism and thus Z ∼ = FG(Z ). Therefore Z ∈ B.  Proposition 3.8. Let F : A  B : G be an equivalence between the full additive subcategories A and B of A and B, respectively. The following statements are equivalent: (a) (1) B is closed under θ -faithful factors; (2) A = Genφ ∩ F−1 (B); (b) (1) A is closed with respect to factors modulo φ-generated subobjects; f

g

(2) If 0 → K −→ M −→ N → 0 is an exact sequence in A, with M ∈ A, then F(g) is an epimorphism if and only if K ∈ Genφ . Proof. Since F : A  B : G is an equivalence, we have F(A) ⊆ B ⊆ AdstatF and G(B) ⊆ A ⊆ StatF . (a)⇒(b) (1) It follows by Proposition 3.7. f

g

(2) Let 0 → K −→ M −→ N → 0 be an exact sequence in A, with M ∈ A. If K ∈ Genφ , it follows, by Proposition 3.7, that F(g) is an epimorphism.

F( f )

Conversely, assume that F(g) is an epimorphism. Then, the sequence 0 → F(K ) −→ F(M) GF( f )

F(g)

GF(g)

−→ F(N ) → 0 is exact in B, hence GF(K ) −→ GF(M) −→ GF(N ) → 0 is also exact. Applying the Snake lemma to the following commutative diagram with exact rows GF( f )

GF(g)

GF(K ) −−−−→ GF(M) −−−−→ GF(N ) −−−−→ 0      φ φK  φM   N 0 −−−−→

K

f

−−−−→

M

g

−−−−→

N

−−−−→ 0

and taking into account that M ∈ StatF , we obtain that φ N is an epimorphism, and thus N ∈ Genφ . Moreover, F(N ) ∼ = F(M)/F(K ), F(N ) ∈ Faithθ , F(M) ∈ B and B is closed under θ faithful factors, hence F(N ) ∈ B. Thus N ∈ A and, by Lemma 2.8, we have K ∈ Genφ . (b)⇒(a) (1) It follows from Proposition 3.7. (2) Let M ∈ A. Then M ∈ StatF , hence M ∈ Genφ . On the other hand, F(M) ∈ B, hence M ∈ F−1 (B). Thus M ∈ Genφ ∩ F−1 (B) and therefore A ⊆ Genφ ∩ F−1 (B). For the reverse inclusion, let M ∈ Genφ ∩ F−1 (B). Since F(M) ∈ B, we have GF(M) ∈ A, hence GF(M) ∈ StatF . Since M ∈ Genφ , the morphism φ M : GF(M) −→ M is an epimorφM

phism, hence the sequence 0 → Ker(φ M ) −→ GF(M) −→ M → 0 is exact. From the equality F(φ M ) ◦ θF(M) = 1F(M) , we obtain that F(φ M ) is an epimorphism hence, by (b)(2), Ker(φ M ) ∈ Genφ . By Lemma 2.8, we have M ∈ StatF , and thus GF(M) ∼ = M. It follows that M ∈ A. Therefore Genφ ∩ F−1 (B) ⊆ A.  4. Applications Let U ∈ StatF with F(U ) = V . If we set B0 = add(V ) we have, by Lemma 2.5, that B0 ⊆ AdstatF and A0 = add(U ). It is easy to show that B = gen(V ) ∩ Faithθ . Moreover, in this setting, we can see that A = {M/K | M ∈ add(U ), K ∈ Genφ }.

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Example 4.1. (1) With the settings presented in Example 3.5, we could consider U to be the right R-module Q. Then V is the right S-module S. It follows that B0 = add(S), A0 = add(Q), B = gen(S) ∩ Faithθ and A = {M/K | M ∈ add(Q), K ∈ Genφ }; (2) Using the settings from Example 3.6, we have B0 = add(V ), A0 = add(U ), B = gen(V ) ∩ Faithθ and A = {M/K | M ∈ add(U ), K ∈ Genφ }. Corollary 4.2. Let U ∈ StatF with F(U ) = V . Let B ⋆ = gen(V ) ∩ Faithθ and let A⋆ = {M/K | M ∈ add(U ), K ∈ Genφ }. Then the following statements are equivalent: (a) F : A⋆  B ⋆ : G is an equivalence and B ⋆ is closed under θ-faithful factors; (b) B ⋆ ⊆ AdstatF ; f

g

(c) If 0 → K −→ M −→ N → 0 is an exact sequence in A with M ∈ add(U ) and K ∈ Genφ , then F(g) is an epimorphism; f

g

(d) If 0 → K −→ U n −→ N → 0 is an exact sequence in A with K ∈ Genφ , then F(g) is an epimorphism. Proof. From Theorem 3.3, we have (a)⇔(b)⇔(c). (c)⇒(d) It is obvious. f

g

(d)⇒(c) Let 0 → K −→ M −→ N → 0 be an exact sequence in A, with M ∈ add(U ) and K ∈ Genφ . Since M ∈ add(U ), we have M ⊕ M ′ ∼ = U n , for some object M ′ ∈ A and for some positive integer n. If we denote by p the projection on M, then the composition gp gp is an epimorphism, hence we have the exact sequence 0 → Ker(gp) −→ U n −→ N → 0. We observe that Ker(gp) ∼ = K ⊕ M ′ . From the facts that K ∈ Genφ and M ′ ∈ add(U ), hence ′ M ∈ StatF ⊆ Genφ , we obtain that Ker(gp) ∈ Genφ . Applying (d) to the second sequence, we obtain that F(gp) is an epimorphism, hence F(g) ◦ F( p) is an epimorphism and thus F(g) is an epimorphism.  Corollary 4.3. Let A and B be full additive subcategories of A and B, respectively. Let U ∈ A with F(U ) = V . Assume that V k ∈ B, for all positive integers k. Let B ⋆ = gen(V ) ∩ Faithθ and let A⋆ = {M/K | M ∈ add(U ), K ∈ Genφ }. If F : A  B : G is an equivalence with B closed under θ-faithful factors, then F : A⋆  B ⋆ : G is an equivalence with B ⋆ closed under θ -faithful factors. Proof. Suppose that F : A  B : G is an equivalence and B is closed under θ -faithful factors. Then B ⊆ AdstatF and, since A ⊆ StatF , we have that U ∈ StatF . Let Z ∈ B ⋆ . Since Z ∈ gen(V ), there exists an epimorphism g : V n → Z , for some positive integer n. Given that Z ∼ = V n /Ker(g), V n ∈ B and B is closed under θ -faithful factors, we obtain that Z ∈ B, hence Z ∈ AdstatF . Thus B ⋆ ⊆ AdstatF . Now, the conclusion follows by Corollary 4.2 ((a)⇔(b)).  For the next results, we assume that the right derived functors of F does exist. For example, we could consider that the category A has enough injectives or is a Grothendieck category. We denote by R j F the j-th right derived functor of F. We also consider the perpendicular class ⊥ A of all objects M ∈ A for which R1 F(M) = 0. Corollary 4.4. Let A and B be full additive subcategories of A and B, respectively. Let U ∈ A. Assume that R1 F(U ) = 0 and U k ∈ A, for all positive integers k. If F : A  B : G is an equivalence with the class B closed under θ -faithful factors, then cog(U ) ∩ Genφ ⊆⊥ A.

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Proof. Suppose that F : A  B : G is an equivalence and B is closed under θ -faithful factors. Let K ∈ cog(U ) ∩ Genφ . Since K ∈ cog(U ), there is a monomorphism f : K → U n , for some f

p

positive integer n. Hence, we have the exact sequence 0 → K −→ U n −→ U n /K → 0 with K ∈ Genφ and U n ∈ A. By Proposition 3.7, we obtain that F( p) is an epimorphism. On the other hand, we have the following long exact sequence F( f )

∆0

F( p)

0 → F(K ) −→ F(U n ) −→ F(U n /K ) −→ ∆0

R1 F( f )

R1 F( p)

∆1

−→ R1 F(K ) −→ R1 F(U n ) −→ R1 F(U n /K ) −→ · · · . From the facts that F( p) is an epimorphism and R1 F(U ) = 0, it follows that R1 F(K ) = 0, namely K ∈⊥ A. Therefore cog(U ) ∩ Genφ ⊆⊥ A.  Proposition 4.5. Let U ∈ StatF with F(U ) = V . Assume that R1 F(U ) = 0. Let B ⋆ = gen(V )∩Faithθ and let A⋆ = {M/K | M ∈ add(U ), K ∈ Genφ }. Then the following statements are equivalent: (a) F : A⋆  B ⋆ : G is an equivalence and B ⋆ is closed under θ -faithful factors; (b) B ⋆ ⊆ AdstatF ; (c) cog(U ) ∩ Genφ ⊆⊥ A. Proof. (a)⇔(b) It follows by applying Theorem 3.3. (a)⇒(c) It is a consequence of Corollary 4.4. (c)⇒(b) Assume that cog(U ) ∩ Genφ ⊆⊥ A. Let Z ∈ B ⋆ . Since Z ∈ gen(V ), there is an g

i

g

epimorphism V n −→ Z → 0, for some positive integer n. Then 0 → Ker(g) −→ V n −→ σ

G(g)

Z → 0 is an exact sequence, hence the sequence 0 → KerG(g) −→ G(V n ) −→ G(Z ) → 0 is also exact, where σ comes from the canonical decomposition G(i) = σ ◦ π. By Corollary 2.6, KerG(g) ∈ Genφ . Since G(V n ) ∼ = U n , we have KerG(g) ∈ cog(U ). It follows that KerG(g) ∈ cog(U ) ∩ Genφ , hence KerG(g) ∈⊥ A, i.e. R1 F(KerG(g)) = 0. It follows that the sequence F(σ )

FG(g)

0 → F(KerG(g)) −→ FG(V n ) −→ FG(Z ) → 0 is exact, hence FG(g) is an epimorphism. In the following commutative diagram Vn   θV n 

g

−−−−→

Z  θ Z

FG(g)

FG(V n ) −−−−→ FG(Z ) FG(g) is an epimorphism and θV n is an isomorphism. It follows that θ Z is an epimorphism. But θ Z is also a monomorphism, hence θ Z is an isomorphism, namely Z ∈ AdstatF . Therefore B ⋆ ⊆ AdstatF .  Acknowledgments I would like to mention that this paper is part of my Ph.D. Thesis, supervised by Professor Andrei Marcus. The author is supported by the UEFISCSU (CNCSIS) grants ID 489 and BD 166.

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References [1] U. Albrecht, S. Breaz, W. Wickless, The finite quasi-Baer property, J. Algebra 293 (1) (2005) 1–16. [2] U. Albrecht, S. Breaz, W. Wickless, A-solvability and mixed abelian groups, Comm. Algebra 37 (2) (2009) 439–452. [3] U. Albrecht, S. Breaz, W. Wickless, S ∗ -groups, J. Algebra Appl. 10 (2) (2011) 357–363. [4] D.M. Arnold, Endomorphism rings and subgroups of finite rank torsion-free abelian groups, Rocky Mountain J. Math. 12 (1982) 241–256. [5] S. Bazzoni, Equivalences induced by infinitely generated tilting modules, Proc. Amer. Math. Soc. 138 (2) (2010) 533–544. [6] S. Breaz, Almost-flat modules, Czechoslovak Math. J. 53 (2) (2003) 479–489. [7] S. Breaz, The quasi-Baer-splitting property for mixed Abelian groups, J. Pure Appl. Algebra 191 (1–2) (2004) 75–87. [8] S. Breaz, A Morita type theorem for a sort of quotient categories, Czechoslovak Math. J. 55 (1) (2005) 133–144. [9] S. Breaz, C. Modoi, F. Pop, Natural equivalences and dualities, in: Proceedings of the International Conference on Modules and Representation Theory, Cluj-Napoca, July 7–12, 2008, Presa Universitarˇa Clujeanˇa, Cluj-Napoca, 2009, pp. 25–40. [10] F. Casta˜no-Iglesias, On a natural duality between Grothendieck categories, Comm. Algebra 36 (6) (2008) 2079–2091. [11] F. Casta˜no-Iglesias, J. G´omez-Torrecillas, R. Wisbauer, Adjoint functors and equivalences of subcategories, Bull. Sci. Math. 127 (5) (2003) 379–395. [12] R.R. Colby, K.R. Fuller, Equivalence and Duality for Module Categories, in: Cambridge Tracts in Math., vol. 161, Cambridge University Press, Cambridge, 2004. [13] R. Colpi, Tilting in Grothendieck categories, Forum Math. 11 (6) (1999) 735–759. [14] R. Colpi, K.R. Fuller, Tilting objects in abelian categories and quasitilted rings, Trans. Amer. Math. Soc. 359 (2) (2007) 741–765. [15] K.R. Fuller, Density and equivalence, J. Algebra 29 (3) (1974) 528–550. [16] K.R. Fuller, Natural and doubly natural dualities, Comm. Algebra 34 (2) (2006) 749–762. [17] P. Gabriel, Des categories abeliennes, Bull. Soc. Math. France 90 (1962) 323–448. [18] E. Gregorio, Tilting equivalences for Grothendieck categories, J. Algebra 232 (2) (2000) 541–563. [19] F. Mantese, A. Tonolo, Natural dualities, Algebr. Represent. Theory 7 (2004) 43–52. [20] A. Marcus, C. Modoi, Graded endomorphism rings and equivalences, Comm. Algebra 31 (7) (2003) 3219–3249. [21] C. Menini, A. Orsatti, Representable equivalences between categories of modules and applications, Rend. Semin. Mat. Univ. Padova 82 (1989) 203–231. [22] C. N˘ast˘asescu, F. Van Oystaeyen, Methods of Graded Rings, in: Lecture Notes in Math., vol. 1836, Springer, Berlin, 2004. [23] F. Pop, Natural dualities between abelian categories, Cent. Eur. J. Math. 9 (5) (2011) 1088–1099. [24] W. Rump, ∗-modules, tilting, and almost abelian categories, Comm. Algebra 29 (8) (2001) 3293–3325. [25] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia, 1991. [26] R. Wisbauer, Tilting in module categories, in: Abelian Groups, Module Theory, and Topology (Padova, 1997), in: Lect. Notes Pure Appl. Math., vol. 201, Marcel Dekker, New York, 1998, pp. 421–444.