Abstract

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:
 
 The functors \overline{EXT}^n_{Comp(A-Mod)}(X,-): Comp(A-Mod) \longrightarrow Comp(B-Mod) and Tor_n^{Comp(B-Mod)}(X,-): Comp(B-Mod) \longrightarrow Comp(A-Mod) are adjoint functors.
 The  functor S_C^{-1}() commute with  the functors X\bigotimes - , Hom^{\bullet}(X,-) and their corresponding derived functors  \overline{EXT}^n_{Comp(A-Mod)}(X,-) and  Tor_n^{Comp(B-Mod)}(X,-).

Highlights

  • The aim of this paper is to show that: 1. The functors EXT Cn omp(A−Mod)(X, −) : Comp(A − Mod) −→ Comp(B − Mod) and T ornComp(B−Mod)(X, −) : Comp(B − Mod) −→ Comp(A − Mod) are adjoint functors

  • 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), theorem 2.76 for instance)

  • −), where n is considered to be the n-th funtor derived of Hom, are isomorphic and on the other hand X − and T or0Comp(B−Mod)(X, −), where T ornComp(B−Mod) is the n-th derived functor of the tensor product functor X −, are isomorphic we can conclude that EXT C0 omp(A−Mod)(X, −) and

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Summary

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), theorem 2.76 for instance). The adjunction study between Hom functor and tensor product functor has been done by several authors in the category A − Mod of A-modules That is the functors HomA(M, −) and M − ,where M is a (A − B) bimodule, are adjoint functors. Its analogue, considered in the category of complexes, has been shown in Otherwise the functors Hom(X, −) and X − are adjoint functors, where X is a complex of (A − B) bimodules

Now since on the one hand
Otherwise if the functors n C
Journal of Mathematics Research
Since is a projective resolution of then
Let be
At first we have
According to theorem
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