We generalize r-costar module to r-costar pair of contravariant functors between abelian categories.

1. Introduction

For a ring A, a fixed right A-module M, and D=EndA(M), let fgd-tl(MA) denote the class of torsionless right R-modules whose M-dual are finitely generated over D and fg-tl (MD) denote the class of finitely generated torsionless left D-modules. M is called costar module if
(1.1)HomA(-,M):fgd-tl(MA)⇄fg-tl(MD):HomD(-,M)
is a duality. Costar modules were introduced by Colby and Fuller in [1]. M is said to be an r-costar module provided that any exact sequence
(1.2)0→X→Y→Z→0,
such that X and Y are M-reflexive, remains exact after applying the functor HomA(-,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
(1.3)0→X→Mk0→Mk1→⋯→Mkn-1
such that ki 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-self-cotilting module provided that any exact sequence
(1.4)0→X→Mm→Z→0,
such that Z∈n-cop(M) and m is a positive integer, remains exact after applying the functor HomA(-,M) and n-cop(Q)=(n+1)-cop(Q). Finitistic n-self-cotilting modules were introduced by Breaz in [3].

In [4] Castaño-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.

2. Preliminaries

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
(2.1)ηX,Y:Homℭ(X,G(Y))→Hom𝔇(Y,F(X)),
for X∈ℭ and Y∈𝔇. Then they induce two natural transformations δ:1ℭ→GF and δ′:1𝔇→FG defined by δX=ηX,F(X)-1(1F(X)) and δY′=ηG(Y),Y-1(1G(Y)). Moreover the following identities are satisfied for each X∈ℭ and Y∈𝔇:
(2.2)F(δX)∘δF(X)′=1F(X),G(δY′)∘δG(Y)′=1G(Y).
The pair (F,G) is called a duality if there are functorial isomorphisms GF≃1ℭ and FG≃1𝔇. An object X of ℭ (Y∈𝔇) is called F-reflexive (resp., G-reflexive) in case δX (resp., δY′) 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).

We say that the pair (F,G) of left exact contravariant functors is r-costar provided that any exact sequence
(2.3)0→Q→U→V→0,
with Q,U∈Ref(F) remains exact after applying the functor F if and only if V∈Ref(F).

An object U is called V-finitely generated if there is an epimorphism Vn→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 X(2.4)P:⋯→P2→P1→P0→X→0.
This gives rise to the sequence
(2.5)0→G(X)→G(P0)→G(P1)→G(P2)→⋯,
and the cochain complex G(P), which we can compute its cohomology at the nth spot (the kernel of the map from G(Pn) modulo the image of the map to G(Pn)) and denote it by Hn(G(P)). We define RnG(X)=Hn(G(P)) as the nth right derived functor of G. For the functor Gwe define T⊥Gi⩾n={X∈𝔇:RiG(X)=0 for every i⩾n}.

Let
(2.6)0→Q→fU→V→0
be an exact sequence in ℭ. Applying the functor F we get the exact sequence
(2.7)0→F(V)→F(U)→pX→0,
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
(2.8)0→GX→G(p)GF(U)→GF(V).
Now if we put α=G(j)∘δQ, then
(2.9)G(p)∘α=G(p)∘G(j)∘δQ=G(j∘p)∘δQ=GF(f)∘δQ=δU∘f.
So we have the following commutative diagram
(2.10)

3. r-Costar Pair of Contravariant Functors

We will fix all the notations and terminologies used in previous section.

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⊥Gi⩾1 and T⊥Gi⩾0=0. Then (F,G) is an r-costar.

Proof.

Let
(3.1)0→Q→fU→V→0
be an exact sequence with Q,U∈Ref(F). Assume that we have the exact sequence
(3.2)0→F(V)→F(U)→F(Q)→0,
after applying the functor F. Applying the functor G to the last sequence, we get an exact sequence
(3.3)0→GF(Q)→GF(U)→GF(V)→R1G(F(Q))=0,
since F(Q)∈Ref(G)⊆T⊥Gi⩾1. Hence we have the following commutative diagram:
(3.4)
Since Q,U∈Ref(F), δQ and δU are isomorphisms. Now it is clear that δV is an isomorphism which means that V∈Ref(F).

Conversely, suppose that V∈Ref(F). Applying the functor F to the sequence (3.1), we get an exact sequence
(3.5)0→F(V)→F(U)→X→0,
where X=Im(F(f)). Hence we can get the exact sequence
(3.6)0→X→jF(Q)→Y→0,
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))
(3.7)
where α=G(j)∘δQ. Note that δU and δV are isomorphisms, since Q,U∈Ref(F). lt is clear from the diagram that R1G(X)=0. Now RiG(X)=0 for all i⩾2, by dimension shifting, since F(U),F(V)∈Ref(G)⊆T⊥Gi⩾1. Hence X∈T⊥Gi⩾1. Now applying the functor G to sequence (3.6), we get the long exact sequence
(3.8)0→G(Y)→GF(Q)→G(j)G(X)→R1G(Y)→R1G(F(Q))→R1G(X)→R2G(Y)→R2G(F(Q))→⋯.
Above we conclude that X∈T⊥Gi⩾1 and by assumptions F(Q)∈Ref(G)⊆T⊥Gi⩾1, thus R1G(X)=0 and R2G(F(Q))=0. Hence by dimension shifting Y∈T⊥Gi⩾2. Now consider the following part from sequence (3.8)
(3.9)0→G(Y)→GF(Q)→G(j)G(X)→R1G(Y).
Note that α=G(j)∘δQ in diagram (3.7) is an isomorphism, since δU and δV are isomorphisms. Hence G(j) is an isomorphism, since δQ is an isomorphism, so from sequence (3.9), R1G(Y)=0=G(Y).We conclude that Y∈T⊥Gi⩾0. Since T⊥Gi⩾0=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
(3.10)0→Q→U→V→0
in Ref(F). 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(G)=T⊥Gi⩾1. Then (F,G) is an r-costar.

Proof.

Let X∈T⊥Gi⩾0, then X∈T⊥Gi⩾1=Ref(G). Hence X≅FG(X)=0.

Proposition 3.3.

Let V be a G-reflexive generator in 𝔇 and U=G(V). Let (F,G) be an r-costar pair. If Ref(G)⊆g(V), then for any X∈Ref(F), there is an infinite exact sequence
(3.11)0→X→U1→⋯→Un→⋯,
which remains exact after applying the functor F, where Ui∈add(U) for each i.

Proof.

Let X∈Ref(F). Then F(X)∈Ref(G), so by assumption there is an exact sequence
(3.12)Vn→F(X)→0.
Applying the functor G we have an exact sequence
(3.13)0→X→Un→Y→0,
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
(3.14)0→F(Y)→F(Un)→F(X)→0.
Applying the functor G again we get the following commutative diagram with exact rows
(3.15)
Since X, Un∈Ref(F), Y∈Ref(F). 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 Ref(G)⊆gen(V). Then Ref(G)⊆T⊥Gi⩾1.

Proof.

Let X∈Ref(G), then G(X)∈Ref(F) and hence by Proposition 3.3, there is an infinite exact sequence
(3.16)0→G(X)→U1→⋯→Un→⋯,
which remains exact after applying the functor F, where Ui∈add(U) for each i. So we have an exact sequence
(3.17)⋯→F(Un)→⋯→F(U1)→FG(X)→0
Again the last sequence remains exact after applying the functor G, since we get a sequence isomorphic to sequence (3.16), because G(X), Ui, for each i, are F-reflexive. We obtain that Ref(G)⊆T⊥Gi⩾1 by dimension shifting.

Suppose we have the following exact sequence in 𝔇(3.18)0→X→P2→P1→Y→0,
where P2, P1 are projective objects in 𝔇 and Y∈T⊥Gi⩾1. Applying the functor G we get the following exact sequence
(3.19)0→G(Y)→G(P1)→G(P2)→G(X)→R1G(Y)=0.

Applying the functor F we get the following commutative diagram with exact rows
(3.20)
If proj(𝔇)⊆Ref(G), then it is clear that X∈Ref(G).

Proposition 3.5.

Let V be a G-reflexive generator in 𝔇. Let (F,G) be an r-costar pair and suppose that proj(𝔇)⊆Ref(G)⊆gen(V). Then T⊥Gi⩾0=0.

Proof.

For any X∈T⊥Gi⩾0, we can build the following exact sequence in 𝔇(3.21)0→Y→P2→P1→X→0,
where P2, P1 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 sequence
(3.22)0→G(X)=0→G(P1)→G(P2)→G(Y)→R1G(X)=0.
Applying the functor F we get the following commutative diagram with exact rows
(3.23)
Thus it is clear that X≅0.

Now we are able to give the following characterization of r-costar pair.

Theorem 3.6.

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 proj(𝔇)⊆Ref(G)⊆gen(V). Then (F,G) is an r-costar if and only if Ref(G)⊆T⊥Gi⩾1and T⊥Gi⩾0=0.

Proof.

By Propositions 3.4, 3.1, and 3.5.

Corollary 3.7.

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 U=G(V). If proj(𝔇)⊆Ref(G)⊆gen(V), then the following are equivalent.

(F,G) is an r-costar.

For any exact sequence
(3.24)0→X→Un→Y→0,

with X∈Ref(F), then Y∈Ref(F) if and only if the exact sequence remains exact after applying the functor F.
Proof.

(1)⇒(2) follows from the definition of r-costar pair.

(2)⇒(1) the proof goes the same as the proofs of Propositions 3.3, 3.4, 3.5, and Theorem 3.6.

ColbyR. R.FullerK. R.Costar modulesLiuH.ZhangS.r-costar modulesBreazS.Finitistic n-self-cotilting modulesCastaño-IglesiasF.On a natural duality between Grothendieck categoriesPopF.Natural dualities between abelian categoriesBreazS.PopF.Dualities induced by right adjoint contravariant functors