r-Costar Pair of Contravariant Functors

Abstract

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=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 (_𝐷𝑀) denote the class of finitely generated torsionless left 𝐷-modules. 𝑀 is called costar module if Hom_𝐴(−,𝑀)∶fgd-tl𝑀_𝐴⇄fg-tl_𝐷𝑀∶Hom_𝐷(−,𝑀) (1.1) 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 0→𝑋→𝑌→𝑍→0, (1.2) 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 0→𝑋→𝑀^𝑘0→𝑀^𝑘1→⋯→𝑀^𝑘𝑛−1 (1.3) 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-self-cotilting module provided that any exact sequence 0→𝑋→𝑀^𝑚→𝑍→0, (1.4) 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] 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 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^′∶1_𝔇→𝐹𝐺 defined by 𝛿_𝑋=𝜂⁻¹_{𝑋,𝐹(𝑋)}(1_𝐹(𝑋)) and 𝛿^′_𝑌=𝜂⁻¹_{𝐺(𝑌),𝑌}(1_𝐺(𝑌)). Moreover the following identities are satisfied for each 𝑋∈ℭ and 𝑌∈𝔇: 𝐹𝛿_𝑋∘𝛿^′_𝐹(𝑋)=1_𝐹(𝑋),𝐺𝛿^′_𝑌∘𝛿^′_𝐺(𝑌)=1_𝐺(𝑌). (2.2) The pair (𝐹,𝐺) is called a duality if there are functorial isomorphisms 𝐺𝐹≃1_ℭ and 𝐹G≃1_𝔇. An object 𝑋 of ℭ (𝑌∈𝔇) is called 𝐹-reflexive (resp., 𝐺-reflexive) in case 𝛿_𝑋 (resp., 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).

We say that the pair (𝐹,𝐺) of left exact contravariant functors is r-costar provided that any exact sequence 0→𝑄→𝑈→𝑉→0, (2.3) 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 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 𝑋𝑃:⋯→𝑃₂→𝑃₁→𝑃₀→𝑋→0. (2.4) This gives rise to the sequence 0→𝐺(𝑋)→𝐺𝑃₀→𝐺𝑃₁→𝐺𝑃₂→⋯, (2.5) 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^𝑛𝐺(𝑋)=𝐻^𝑛(𝐺(𝑃)) as the 𝑛th right derived functor of 𝐺. For the functor 𝐺we define ^⟂𝑇_𝐺^𝑖⩾𝑛={𝑋∈𝔇∶𝑅^𝑖𝐺(𝑋)=0 for every 𝑖⩾𝑛}.

Let 0→𝑄−→^𝑓𝑈→𝑉→0 (2.6) be an exact sequence in ℭ. Applying the functor 𝐹 we get the exact sequence 0→𝐹(𝑉)→𝐹(𝑈)−→^𝑝𝑋→0, (2.7) 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 0→𝐺𝑋−−−−^𝐺𝑝→𝐺𝐹(𝑈)→𝐺𝐹(𝑉). (2.8) Now if we put 𝛼=𝐺(𝑗)∘𝛿_𝑄, then 𝐺𝑝∘𝛼=𝐺𝑝∘𝐺𝑗∘𝛿_𝑄=𝐺𝑗∘𝑝∘𝛿_𝑄=𝐺𝐹𝑓∘𝛿_𝑄=𝛿_𝑈∘𝑓. (2.9) 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_𝐺^𝑖⩾1 and ^⟂𝑇_𝐺^𝑖⩾0=0. Then (𝐹,𝐺) is an r-costar.

Proof. Let 0→𝑄−→^𝑓𝑈→𝑉→0 (3.1) be an exact sequence with 𝑄,𝑈∈Ref(𝐹). Assume that we have the exact sequence 0→𝐹(𝑉)→𝐹(𝑈)→𝐹(𝑄)→0, (3.2) after applying the functor 𝐹. Applying the functor 𝐺 to the last sequence, we get an exact sequence 0→𝐺𝐹(𝑄)→𝐺𝐹(𝑈)→𝐺𝐹(𝑉)→𝑅¹𝐺(𝐹(𝑄))=0, (3.3) since 𝐹(𝑄)∈Ref(𝐺)⊆^⟂𝑇_𝐺^𝑖⩾1. Hence we have the following commutative diagram: (3.4) Since 𝑄,𝑈∈Ref(𝐹),  𝛿_𝑄 and 𝛿_𝑈 are isomorphisms. Now it is clear that 𝛿_𝑉 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 0→𝐹(𝑉)→𝐹(𝑈)→𝑋→0, (3.5) where 𝑋=Im(𝐹(𝑓)). Hence we can get the exact sequence 0→𝑋→−^𝑗𝐹(𝑄)→𝑌→0, (3.6) 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)∘𝛿_𝑄. Note that 𝛿_𝑈 and 𝛿_𝑉 are isomorphisms, since 𝑄,𝑈∈Ref(𝐹). lt is clear from the diagram that 𝑅¹𝐺(𝑋)=0. Now 𝑅^𝑖𝐺(𝑋)=0 for all 𝑖⩾2, by dimension shifting, since 𝐹(𝑈),𝐹(𝑉)∈Ref(𝐺)⊆^⟂𝑇_𝐺^𝑖⩾1. Hence 𝑋∈^⟂𝑇_𝐺^𝑖⩾1. Now applying the functor 𝐺 to sequence (3.6), we get the long exact sequence 0→𝐺(𝑌)→𝐺𝐹(𝑄)−−−−→^𝐺𝑗𝐺(𝑋)→𝑅¹𝐺(𝑌)→𝑅¹𝐺(𝐹(𝑄))→𝑅¹𝐺(𝑋)→𝑅²𝐺(𝑌)→𝑅²𝐺(F(𝑄))→⋯. (3.8) Above we conclude that 𝑋∈^⟂𝑇_𝐺^𝑖⩾1 and by assumptions 𝐹(𝑄)∈  Ref(𝐺)⊆^⟂𝑇_𝐺^𝑖⩾1, thus 𝑅¹𝐺(𝑋)=0 and 𝑅²𝐺(𝐹(𝑄))=0. Hence by dimension shifting 𝑌∈^⟂𝑇_𝐺^𝑖⩾2. Now consider the following part from sequence (3.8) 0→G(𝑌)→𝐺𝐹(𝑄)−−−−→^𝐺𝑗𝐺(𝑋)→𝑅¹𝐺(𝑌). (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), 𝑅¹𝐺(𝑌)=0=𝐺(𝑌).We conclude that 𝑌∈^⟂𝑇_𝐺^𝑖⩾0. Since ^⟂𝑇_𝐺^𝑖⩾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 0→𝑄→𝑈→𝑉→0 (3.10) 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_𝐺^𝑖⩾1. Then (𝐹,𝐺) is an r-costar.

Proof. Let 𝑋∈^⟂𝑇_𝐺^𝑖⩾0, then 𝑋∈^⟂𝑇_𝐺^𝑖⩾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 0→𝑋→𝑈₁→⋯→𝑈_𝑛→⋯, (3.11) which remains exact after applying the functor 𝐹, where 𝑈_𝑖∈add(𝑈) for each 𝑖.

Proof. Let 𝑋∈Ref(𝐹). Then 𝐹(𝑋)∈Ref(𝐺), so by assumption there is an exact sequence 𝑉^𝑛→𝐹(𝑋)→0. (3.12) Applying the functor 𝐺 we have an exact sequence 0→𝑋→𝑈^𝑛→𝑌→0, (3.13) 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 0→𝐹(𝑌)→𝐹𝑈^𝑛→𝐹(𝑋)→0. (3.14) Applying the functor 𝐺 again we get the following commutative diagram with exact rows (3.15) Since 𝑋, 𝑈^𝑛∈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_𝐺^𝑖⩾1.

Proof. Let 𝑋∈Ref(𝐺), then 𝐺(𝑋)∈Ref(𝐹) and hence by Proposition 3.3, there is an infinite exact sequence 0→𝐺(𝑋)→𝑈₁→⋯→𝑈_𝑛→⋯, (3.16) which remains exact after applying the functor 𝐹, where 𝑈_𝑖∈add(𝑈) for each 𝑖. So we have an exact sequence ⋯→𝐹𝑈_𝑛→⋯→𝐹𝑈₁→𝐹𝐺(𝑋)→0 (3.17) 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_𝐺^𝑖⩾1 by dimension shifting.

Suppose we have the following exact sequence in 𝔇0→𝑋→𝑃₂→𝑃₁→𝑌→0, (3.18) where 𝑃₂,  𝑃₁ are projective objects in 𝔇 and 𝑌∈^⟂𝑇_𝐺^𝑖⩾1. Applying the functor 𝐺 we get the following exact sequence 0→𝐺(𝑌)→𝐺𝑃₁→𝐺𝑃₂→𝐺(𝑋)→𝑅¹𝐺(𝑌)=0. (3.19)

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_𝐺^𝑖⩾0=0.

Proof. For any 𝑋∈^⟂𝑇_𝐺^𝑖⩾0, we can build the following exact sequence in 𝔇0→𝑌→𝑃₂→𝑃₁→𝑋→0, (3.21) 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 0→𝐺(𝑋)=0→𝐺𝑃₁→𝐺𝑃₂→𝐺(𝑌)→𝑅¹𝐺(𝑋)=0. (3.22) 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_𝐺^𝑖⩾1and ^⟂𝑇_𝐺^𝑖⩾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.(1)(𝐹,𝐺) is an r-costar.(2)For any exact sequence 0→𝑋→𝑈^𝑛→𝑌→0, (3.24)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.