Localization in the Category $COMP(G_{r}(A-Mod))$ of Complex associated to the Category $G_{r}(A-Mod)$ of Graded left $A-$modules over a Graded Ring

Abstract

The main results of this paper are : \\
If $A=\displaystyle{\bigoplus_{n\in\mathbb{Z}}}A_{n}$ is a graded duo-ring, $S_{H}$ is a part
formed of regulars homogeneous elements of $A$, $\overline{S}_{H}$ is the homogeneous multiplicatively
closed subset of $A$
generated by $S_{H}$, then:

\begin{enumerate}
\item The relation $C_{H}(-) :G_{r}(\overline{S}_{H}^{-1}A-Mod)\longrightarrow COMP(G_{r}(\overline{S}_{H}^{-1}A-Mod))$ which that for all graded left
$\overline{S}_{H}^{-1}A-$module $\overline{S}_{H}^{-1}M$ of $G_{r}(\overline{S}_{H}^{-1}A-Mod)$
we correspond the associate complex sequence $(\overline{S}_{H}^{-1}M)_{*}$ to a graded $\overline{S}_{H}^{-1}A-$module
$\overline{S}_{H}^{-1}M$ and for all graded morphism of graded left $\overline{S}_{H}^{-1}A-$modules
$\overline{S}_{H}^{-1}f : \overline{S}_{H}^{-1}M\longrightarrow \overline{S}_{H}^{-1}N$ of degree $k$
we correspond the associated complex chain
$(\overline{S}_{H}^{-1}f)_{*}^{k}$ to a morphism of graded left $\overline{S}_{H}^{-1}A-$module
$\overline{S}_{H}^{-1}f : \overline{S}_{H}^{-1}M\longrightarrow \overline{S}_{H}^{-1}N$
is an additively exact covariant functor.
\item The relation $(C_{H}\circ\overline{S}_{H}^{-1})(-) :G_{r}(A-Mod)\longrightarrow COMP(G_{r}(\overline{S}_{H}^{-1}A-Mod))$ which that for all graded left
$A-$module $M$ of $G_{r}(A-Mod)$
we correspond the associate complex sequence $(C_{H}\circ\overline{S}_{H}^{-1})(M)=(\overline{S}_{H}^{-1}M)_{*}$ to a graded $A-$module
$M$ and for all graded morphism of graded left $A-$modules
$f : M\longrightarrow N$ of degree $k$
we correspond the associated complex chain
$(C_{H}\circ\overline{S}_{H}^{-1})(f)=(\overline{S}_{H}^{-1}f)_{*}^{k}$ to a morphism of graded left $A-$module
$f : M\longrightarrow N$
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:
$$\overline{S}^{-1}_{H}((H_{n}\circ C)(M))\cong H_{n}(C_{H}\circ \overline{S}^{-1}_{H})(M)).$$
\end{enumerate}