On generalized Hartshorne's conjecture and local cohomology modules

TL;DR

This paper investigates the cofiniteness of generalized local cohomology modules and extends Hartshorne's conjecture, providing partial answers and new insights into the structure of these modules and their attached primes.

Contribution

It introduces the concept of $rak{a}$-weakly finite modules and studies their role in the cofiniteness of generalized local cohomology modules, extending prior conjectures.

Findings

Partial results on the cofiniteness of generalized local cohomology modules.

New conditions for cofiniteness involving $rak{a}$-weakly finite modules.

Results on attached primes of top generalized local cohomology modules.

Abstract

Let $\mathfrak{a}$ denote an ideal of a commutative Noetherian ring $R$. Let $M$ and $N$ be two $R$-modules. In this paper, we give partial answers on the extension of Hartshorne's conjecture about the cofiniteness of torsion and extension functors. For this purpose, we study the cofiniteness of the generalized local cohomology module $H^{i}_{\mathfrak{a}}(M, N)$ for a new class of modules, called $\mathfrak{a}$-weakly finite modules, in the local and non-local case. Furthermore, we derive some results on attached primes of top generalized local cohomology modules.