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

Articles in Press

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Science, Payame Noor University (PNU), Tehran, Iran

Abstract

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  $\mathcal{S}_{\frak{a}}^{\frak{b}}(M):=\inf \lbrace i\in \mathbb{N}_{0} : \frak{b}^t\text{H}_{\frak{a}}^{i}(M)  \text{ is not } \mathcal{S} -\text{minimax} \text{ for all } t\in \mathbb{N} \rbrace$. 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 1 over an arbitrary commutative Noetherian ring $R$ and at all levels whenever $\dim R \leq 2$. 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.

Keywords

References

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Algebraic Structures and Their Applications

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Available Online from 31 July 2025

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