Localization, Isomorphisms and Adjoint Isomorphism in the Category Comp(A − Mod)

A and B are considered to be non necessarily commutative rings and X a complex of (A − B) bimodules. The aim of this paper is to show that: 1. The functors EXT n Comp(A−Mod)(X,−) : Comp(A − Mod) −→ Comp(B − Mod) and TorComp(B−Mod) n (X,−) : Comp(B − Mod) −→ Comp(A − Mod) are adjoint functors. 2. The functor S −1 C () commute with the functors X ⊗ − , Hom•(X,−) and their corresponding derived functors EXT n Comp(A−Mod)(X,−) and Tor Comp(B−Mod) n (X,−).


Introduction
The adjunction study between Hom functor and tensor product functor has been done by several authors in the category A − Mod of A-modules (see Rotman, J., J. (1972) A and B be two rings, X a complex of (A − B) bimodules, C a complex of A-modules and n an integer, we organize this work as following: we give some definitions and preliminary results in our first section for reminder.
In our second section we prove the following results: 4. if A is a subring of B, S a saturated multiplicative subset of A and B satisfying the left Ore conditions then: And finally, in the last section, we show the following results:

Definitions and Preliminary Results
Definition and proposition 2.1 The category of complexes of left A-modules is the category denoted by Comp(A − Mod) such that: 1. objects are complexes of left A-modules.
A complex of left A-modules C is a sequence of homomorphisms of left A-modules (C n d n C −→ C n+1 ) n∈Z such that d n+1 • d n = 0, for all n ∈ Z.
2. Morphisms are maps of complexes of left A-modules. Let C and D be two complexes, a map of complexes of left A-modules f : C −→ D is a sequence of homomorphisms of left A-modules ( f n :

Proposition 2.2
Let A be a ring and S a saturated multiplicative subset of A verifying the left Ore conditions. Then the relation: . . is an objet of Comp(A − Mod) then : Then S −1 C () is an exact covariant functor. Proof see (Dembele, B., Maaouia, B.,F., & Sanghare, M. (2020)), proposition 1

Definition and proposition 2.3:
Let X be a complex of (A − B)-bimodules and let be the following correspondance: such that : Then X − is a covariant functor that is right exact. Let X be a complex of (A − B)-bimodules. Let be the following correspondence: and δ HOm • (X,Y) is defined as following: Then HOm • (X, −) is a covariant functor that is left exact.
Proof see [Dembele, B., Maaouia, B.,F., & Sanghare, M. (2020)], definition and proposition 3 Definition 2.5 Let C be a complex of left A-modules and C • a projective resolution of C such us: Then we said that Ker(d n ) is the n − th kernel ofC • and we denote it by K n .

Adjoint Isomorphism Between EXT and T or in Comp(A − Mod)
Definition 3.1 Let C and D be two categories, F : C −→ D and G : D −→ C two functors. It is said that the couple (F, G) is adjoint if for any A ∈ Ob(C) and for any B ∈ Ob(D), there is an isomorphism: so that: a) For any f ∈ Hom C (A , A), the following diagram is commutative: b) For any g ∈ Hom D (B, B ), the following diagram is commutative: Let C be a complex of left A-modules and C • projective resolution of C of n-th kernel Ker(d n ) = K n . Then the functors EXT n+1 On the other hand, by doing the same thing for maps of complexes, we get the result.  Let C and D be two categories, F and G two functors with same variance from C to D. A natural transformation or functorial morphism from F to G is a map Φ : F −→ G so that: • If F and G are contravariant then the following diagram is commutative: If Φ M is an isomorphism for all M then Φ is called functorial isomorphism.
Definition 4.2 1. We say that a complex of left A-modules C is bounded if for | n | large, C n = 0 .
2. We say that a complex of left A-modules C is of finite type if C is bounded and for all n ∈ Z, C n is of finite type .
3. We say that a complex of left A-modules C is of type FP ∞ if it has a projective resolution: with P n is a finite type complex of left A-modules for all n ≥ 0.

Lemma 4.3
Let C be a complex of A-modules and C • a projective resolution of C of n-th kernel Ker(d n ) = K n . Then the functors EXT n+1 As the one of lemma 3.2 Lemma 4.4 Let C be a complex of A-modules and C • a projective resolution of C of n-th kernel Ker(d n ) = K n . Then where T or Comp(S −1 A−Mod) n (S −1 C (X), S −1 C ()) is the n-th left derived functor of S −1 C (X) S −1 C (). Proof As the one of lemma 3.2.

Theorem 4.5
Let B be a ring, A a sub-ring of B, S a suturated multiplicative subset of A and B verifying the left Ore conditions and X a complex of A − B bimodules. Let be the functors 1. for all complex of left B-modules Y we have: Proof we know, according to the proof of theorem 6 in [Dembele, B., Maaouia, B.,F., & Sanghare, M. (2020)], that for all complex of left A modules Y there exist an isomorphism Φ Y : Now it remaind to prove, for all map of complexes f : Y 1 −→ Y 2 , the commutativity of the following diagram: That is for all integer m the following diagram is commutative: .We have on one hand: And on the other hand we have: 1. for all complex of left A-modules Y we have: Then S −1 C Hom • (X, −) and Hom • (S −1 C (X), S −1 C ()) are isomorphic. Proof we know that according to the proof of theorem 7 in [Dembele, B., Maaouia, B.,F., & Sanghare, M. (2020)] that for all complex of left A modules Y there exist an isomorphism Φ X,Y : S −1 C Hom • (X, Y) −→ Hom • (S −1 C (X), S −1 C (Y)) such that: Now let f : Y 1 −→ Y 2 be a map of complexes, let us show the commutativity of the following diagram: That is for all integers m and t the following diagram commutative: And secondly: On one part we have: and other part we have: According to theorem 4.6 S −1 C Hom • (X, −) . That show us that the relation is true for k = 0.
Assume that it is true for all k < n and show that it is true for n.
According to lemma 3.2 we have: By hypothesis we have: