Some results on generalized local cohomology modules

TL;DR

This paper investigates the behavior of generalized local cohomology modules in relation to local cohomology and extension modules within Serre subcategories, providing new results on their Artinianness, cofiniteness, and support properties.

Contribution

It demonstrates how these modules behave similarly at initial points and establishes new finiteness, Artinianness, and cofiniteness results for generalized local cohomology modules.

Findings

Artinianness of certain generalized local cohomology modules

Cofiniteness results for modules $ ext{lc}^{n}_{a}(M, X)$

Finiteness of support and associated primes of these modules

Abstract

Let $R$ be a commutative Noetherian ring with non-zero identity, $\fa$ an ideal of $R$, $M$ a finite $R$--module and $X$ an arbitrary $R$--module. Here, we show that, in the Serre subcategories of the category of $R$--modules, how the generalized local cohomology modules, the ordinary local cohomology modules and the extension modules behave similarly at the initial points. We conclude some Artinianness and cofiniteness results for $\lc^{n}_{\fa}(M, X)$, and some finiteness results for $\Supp_R(\lc^{n}_{\fa}(M, X))$ and $\Ass_R(\lc^{n}_{\fa}(M, X))$.