Localization in the Category COMP(Gr(A−Mod)) of Complex associated to the Category Gr(A−Mod) of Graded left A−modules over a Graded Ring

Abstract

The main results of this paper are : \
If $A={\displaystyle \underset{n\in \mathbb{Z}}{⨁}{A}_n}$ is a graded duo-ring, $S_H$ is a part
formed of regulars homogeneous elements of $A$, ${\bar{S}}_H$ is the homogeneous multiplicatively
closed subset of $A$
generated by $S_H$, then:

\begin{enumerate}
\item The relation $C_H(-):G_r({\bar{S}}_H^{-1}A-Mod)⟶COMP(G_r({\bar{S}}_H^{-1}A-Mod))$ which that for all graded left
${\bar{S}}_H^{-1}A-$module ${\bar{S}}_H^{-1}M$ of $G_r({\bar{S}}_H^{-1}A-Mod)$
we correspond the associate complex sequence $({\bar{S}}_H^{-1}M{)}_{\ast }$ to a graded ${\bar{S}}_H^{-1}A-$module
${\bar{S}}_H^{-1}M$ and for all graded morphism of graded left ${\bar{S}}_H^{-1}A-$modules
${\bar{S}}_H^{-1}f:{\bar{S}}_H^{-1}M⟶{\bar{S}}_H^{-1}N$ of degree $k$
we correspond the associated complex chain
$({\bar{S}}_H^{-1}f{)}_{\ast }^k$ to a morphism of graded left ${\bar{S}}_H^{-1}A-$module
${\bar{S}}_H^{-1}f:{\bar{S}}_H^{-1}M⟶{\bar{S}}_H^{-1}N$
is an additively exact covariant functor.
\item The relation $(C_H\circ {\bar{S}}_H^{-1})(-):G_r(A-Mod)⟶COMP(G_r({\bar{S}}_H^{-1}A-Mod))$ which that for all graded left
$A-$module $M$ of $G_r(A-Mod)$
we correspond the associate complex sequence $(C_H\circ {\bar{S}}_H^{-1})(M)=({\bar{S}}_H^{-1}M{)}_{\ast }$ to a graded $A-$module
$M$ and for all graded morphism of graded left $A-$modules
$f:M⟶N$ of degree $k$
we correspond the associated complex chain
$(C_H\circ {\bar{S}}_H^{-1})(f)=({\bar{S}}_H^{-1}f{)}_{\ast }^k$ to a morphism of graded left $A-$module
$f:M⟶N$
is an additively exact covariant functor.

\item \noindent For all $n\in \mathbb{Z}$ fixed and for all $M\in G_r(A-Mod)$ we have:
$${\bar{S}}_H^{-1}((H_n\circ C)(M))\cong H_n(C_H\circ {\bar{S}}_H^{-1})(M)).$$
\end{enumerate}