Upper bounds for finiteness of generalized local cohomology modules

TL;DR

This paper investigates upper bounds for the finiteness of generalized local cohomology modules over Noetherian rings, characterizing their membership in Serre subcategories and introducing a generalized cohomological dimension.

Contribution

It introduces a new framework for understanding the finiteness properties of generalized local cohomology modules and extends the concept of cohomological dimension in this context.

Findings

Characterization of module membership in Serre subcategories based on upper bounds.

Introduction of a generalized cohomological dimension for local cohomology modules.

Results on the behavior of modules under ideal quotients and their membership in Serre subcategories.

Abstract

Let $R$ be a commutative Noetherian ring with non-zero identity and $\fa$ an ideal of $R$. Let $M$ be a finite $R$--module of of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the generalized local cohomology modules $\lc^{i}_{\fa}(M,N)$ in certain Serre subcategories of the category of modules from upper bounds. We define and study the properties of a generalization of cohomological dimension of generalized local cohomology modules. Let $\mathcal S$ be a Serre subcategory of the category of $R$--modules and $n \geqslant \pd M$ be an integer such that $\lc^{i}_{\fa}(M,N)$ belongs to $\mathcal S$ for all $i> n$. If $\fb$ is an ideal of $R$ such that $\lc^{n}_{\fa}(M,N/{\fb}N)$ belongs to $\mathcal S$, It is also shown that the module $\lc^{n}_{\fa}(M,N)/{\fb}\lc^{n}_{\fa}(M,N)$ belongs to $\mathcal S$.