\mathcal{S}-minimaxness and local-global principle of local cohomology Modules

Let R be a commutative Noetherian ring, Let \mathcal{S} be a Serre subcategory of the category of R-modules, M a finitely generated R-module and \frak{a}, \frak{b}  two ideals of R such that \frak{b}\subseteq\frak{a}. By using the concept of \mathcal{S}-minimax modules w ve define \mathcal{S}^{\frak{b}}-minimaxness dimension \mathcal{S}_{\frak{a}}^{\frak{b}}(M) of M relative to \frak{a} by  \mathcal{S}_{\frak{a}}^{\frak{b}}(M):=\inf \lbrace i\in \mathbb{N}_{۰} : \frak{b}^t\text{H}_{\frak{a}}^{i}(M)  \text{ is not } \mathcal{S} -\text{minimax} \text{ for all } t\in \mathbb{N} \rbrace. Also, we say that the local global principle for the \mathcal{S}-minimaxness of local cohomology modules holds at level r if, for every choice of ideals \frak{a}, \frak{b} of R with \frak{b}\subseteq \frak{a} and for every choice of finitely generated R-module M, it is the case that \mathcal{S}_{\frak{a}}^{\frak{b}}(M)>r\Leftrightarrow f_{\frak{a} R_{\frak{p}}}^{\frak{b} R_{\frak{p}}}(M_{\frak{p}})>r \text{ for all } {\frak{p}} \in \lbrace {\frak{p}} \in\text{Spec}(R)\vert R/{\frak{p}} \notin \mathcal{S}\rbrace. In this paper, we investigate the local-global principle concerning the \mathcal{S}-minimaxness of local cohomology modules. Among other things, we will show that this principle holds at level ۱ over an arbitrary commutative Noetherian ring R and at all levels whenever \dim R \leq ۲. Then by using the obtained results for some specific Serre classes of R-modules we get some main results concerning the local global principle of local cohomology modules.

کلیدواژه ها:

Local cohomology ، Serre subcategory ، \mathcal{S}-minimax modules