r-Costar Pair of Contravariant Functors

S. Al-Nofayee Department of Mathematics, Taif University, P.O. Box 439, Hawiah 21974, Saudi Arabia Correspondence should be addressed to S. Al-Nofayee, alnofayee@hotmail.com Received 8 June 2012; Accepted 16 July 2012 Academic Editor: Feng Qi Copyright q 2012 S. Al-Nofayee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We generalize r-costarmodule to r-costar pair of contravariant functors between abelian categories.


Introduction
M is said to be an r-costar module provided that any exact sequence such that X and Y are M-reflexive, remains exact after applying the functor Hom A −, M if and only if Z is M-reflexive.The notion of r-costar module was introduced by Liu and Zhang in 2 .We say that a right A-module X is n-finitely M-copresented if there exists a long exact sequence such that k i are positive integers for 0 ≤ i ≤ n − 1.The class of all n-finitely M-copresented modules is denoted by n-cop M .We say that a right A-module M is a finitistic n-selfcotilting module provided that any exact sequence such that Z ∈ n-cop M and m is a positive integer, remains exact after applying the functor Hom A −, M and n-cop Q n 1 -cop Q .Finitistic n-self-cotilting modules were introduced by Breaz in 3 .
In 4 Casta ño-Iglesias generalizes the notion of costar module to Grothendieck categories.Pop in 5 generalizes the notion of finitistic n-self-cotilting module to finitistic n-F-cotilting object in abelian categories and he describes a family of dualities between abelian categories.Breaz and Pop in 6 generalize a duality exhibited in 3, Theorem 2.8 to abelian categories.
In this work we continue this kind of study and generalizes the notion of r-costar module to r-costar pair of contravariant functors between abelian categories, by generalizing the work in 2 .We use the same technique of proofs of that paper.

Preliminaries
Moreover the following identities are satisfied for each X ∈ C and Y ∈ D: The pair F, G is called a duality if there are functorial isomorphisms GF We say that the pair F, G of left exact contravariant functors is r-costar provided that any exact sequence An object U is called V -finitely generated if there is an epimorphism V n → X → 0, for some positive integer n.We denote by gen V the subcategory of all V -finitely generated objects.add V denotes the class of all summands of finite direct sums of copies of V .We will denote by proj D the full subcategory of all projective objects in D.
From now on we suppose that D has enough projectives that is, for every object X ∈ D there is a projective object P ∈ D and an epimorphism P → X → 0. It is clear that we can construct a projective resolution for any object X. Suppose we have a projective resolution of This gives rise to the sequence and the cochain complex G P , which we can compute its cohomology at the nth spot the kernel of the map from G P n modulo the image of the map to G P n and denote it by H n G P .We define R n G X H n G P as the nth right derived functor of G.For the functor G we define be an exact sequence in C. Applying the functor F we get the exact sequence where X Im F f .Let F f j • p be the canonical decomposition of F f , where j: X → F Q is the inclusion map.Applying the functor G to the sequence 2.7 , we have the following exact sequence So we have the following commutative diagram International Journal of Mathematics and Mathematical Sciences

r-Costar Pair of Contravariant Functors
We will fix all the notations and terminologies used in previous section.
after applying the functor F. Applying the functor G to the last sequence, we get an exact sequence G .Hence we have the following commutative diagram: Applying the functor F to the sequence 3.1 , we get an exact sequence 0 where X Im F f .Hence we can get the exact sequence for some Y ∈ D, and j is the inclusion map.Applying the functor G to the sequence 3.5 , we have the following exact commutative diagram see diagram 2.10 International Journal of Mathematics and Mathematical Sciences 5 where α G j • δ Q .Note that δ U and δ V are isomorphisms, since G .Now applying the functor G to sequence 3.6 , we get the long exact sequence G .Now consider the following part from sequence 3.8 3.9 Note that α G j • δ Q in diagram 3.7 is an isomorphism, since δ U and δ V are isomorphisms.Hence G j is an isomorphism, since δ Q is an isomorphism, so from sequence 3.9 , 0 by assumptions, Y 0 and hence from sequence 3.6 X ∼ F Q canonically.Therefore the functor F preserves the exactness of the exact sequence We conclude that the pair F, G is an r-costar.

Corollary 3.2. Let F, G be a pair of left exact contravariant functors which are adjoint on the right. Assume that Ref
which remains exact after applying the functor F, where U i ∈ add U for each i.
so by assumption there is an exact sequence Applying the functor G we have an exact sequence International Journal of Mathematics and Mathematical Sciences for some Y ∈ C. Since F, G is an r-costar pair, the last sequence is exact after applying the functor F, that is we have an exact sequence Applying the functor G again we get the following commutative diagram with exact rows By repeating the process to Y , and so on, we finally obtain the desired exact sequence.
Proposition 3.4.Let V be a G-reflexive and a projective generator in D and U G V .Let F, G be an r-costar pair and suppose that which remains exact after applying the functor F, where U i ∈ add U for each i.So we have an exact sequence

3.19
Applying the functor F we get the following commutative diagram with exact rows For a ring A, a fixed right A-module M, and D End A M , let fgd-tl M A denote the class of torsionless right R-modules whose M-dual are finitely generated over D and fg-tl D M denote the class of finitely generated torsionless left D-modules.M is called costar module if Hom A −, M : fgd-tl M A fg-tl D M : Hom D −, M 1.1 is a duality.Costar modules were introduced by Colby and Fuller in 1 .
Let F: C → D and G: D → C be additive and contravariant left exact functors between two abelian categories C and D. It is said that they are adjoint on the right if there are natural isomorphisms η X,Y : Hom C X, G Y −→ Hom D Y, F X , 2.1 for X ∈ C and Y ∈ D. Then they induce two natural transformations δ : 1 C → GF and δ

Proposition 3 . 1 .
Let F, G be a pair of left exact contravariant functors which are adjoint on the right.Assume that Ref G ⊆ ⊥ T i 1

1 G 1 G
Again the last sequence remains exact after applying the functor G, since we get a sequence isomorphic to sequence 3.16 , because G X , U i , for each i, are F-reflexive.We obtain that Ref G ⊆ ⊥ T i by dimension shifting.Suppose we have the following exact sequence inD 0 −→ X −→ P 2 −→ P 1 −→ Y −→ 0, 3.18where P 2 , P 1 are projective objects in D and Y ∈ ⊥ T i .Applying the functor G we get the following exact sequence 0

Proposition 3 . 5 . 0 GTheorem 3 . 6 . 1 GCorollary 3 . 7 .2
Let V be a G-reflexive generator in D. Let F, G be an r-costar pair and suppose that projD ⊆ Ref G ⊆ gen V .Then ⊥ T i 0 G 0.Proof.For any X ∈ ⊥ T i , we can build the following exact sequence in D0 −→ Y −→ P 2 −→ P 1 −→ X −→ 0, 3.21 where P 2 , P 1 are projective objects and Y an object in D. By the argument before the proposition it is clear that Y ∈ Ref G and hence G Y ∈ Ref F .Applying the functor G we get the following exact sequence0 −→ G X 0 −→ G P 1 −→ G P 2 −→ G Y −→ R 1 G X 0. 3.22Applying the functor F we get the following commutative diagram with exact rows it is clear that X ∼ 0.Now we are able to give the following characterization of r-costar pair.Let F, G be a pair of left exact contravariant functor which are adjoint on the right.Suppose that V be a G-reflexive projective generator in D and projD ⊆ Ref G ⊆ gen V .Then F, G is an r-costar if and only if Ref G ⊆ ⊥ T i Let F, Gbe a pair of left exact contravariant functor which are adjoint on the right.Suppose that V be a G-reflexive projective generator in D and U G V .If proj D ⊆ Ref G ⊆ gen V , then the following are equivalent. 1 F, G is an r-costar.For any exact sequence 0 −→ X −→ U n −→ Y −→ 0, 3.24 δ Y is an isomorphism.By Ref F we will denote the full subcategory of all F-reflexive objects.As well by Ref G we will denote the full subcategory of all G-reflexive objects.It is clear that the functors F and G induce a duality between the categories Ref F and Ref G .