Extension functors of generalized local cohomology modules

TL;DR

This paper investigates the properties of generalized local cohomology modules over Noetherian rings, including their membership in Serre subcategories, bounds on injective dimension, cofiniteness, and associated primes.

Contribution

It provides new bounds and criteria for the structure and finiteness properties of generalized local cohomology modules, extending existing theory.

Findings

Upper bounds for injective dimension and Bass numbers of cohomology modules

Results on cofiniteness and minimaxness of the modules

Finiteness of associated primes of the modules

Abstract

Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ an ideal of $R$, $M$ a finitely generated $R$--module, and $X$ an arbitrary $R$--module. In this paper, for non-negative integers $s, t$ and a finitely generated $R$--module $N$, we study the membership of $\operatorname{Ext}_{R}^{s}(N, \operatorname{H}^{t}_{\mathfrak{a}}(M, X))$ in Serre subcategories of the category of $R$--modules and present some upper bounds for the injective dimension and the Bass numbers of $\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$. We also give some results on cofiniteness and minimaxness of $\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$ and finiteness of $\operatorname{Ass}_R(\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$.