Colocalization of formal local cohomology modules

Abstract

Let $(R,\frak{m})$ be a Noetherian local ring, $\frak{a}$ an ideal of $R$ and $M$ a finitely generated $R$-module. In this paper, we study Colocalization of formal local cohomology modules. Here, similar to the local global Principle in local cohomology theory, we investigate artinianness and minimaxness of formal local cohomology modules in terms of their colocalizations. Among other things, we will prove that, for any integer $n$, $\mathfrak{F}_{\frak{a}}^i(M)$ is artinian $R$-module for all $i \lt n$, if and only if $_{\frak{p}}(\mathfrak{F}_{\frak{a}}^i(M)) $ is representable $R_{\frak{p}}$-module for all $i \lt n$ and all $\frak{p} \in \operatorname{Spec}(R)$. Also, $ \mathfrak{F}_{\frak{a}}^i(M) $ is minimax $R$-module for all $i \lt n$, if and only if $ _{\frak{p}}(\mathfrak{F}_{\frak{a}}^i(M)) $ is representable $R_{\frak{p}}$-module for all $i \lt n$ and all $\frak{p} \in \operatorname{Spec}(R)\setminus\lbrace \frak{m}\rbrace$.