Stable Under Specialization Sets and Cofiniteness

TL;DR

This paper introduces and studies a new notion of ofinitenessased on stable under specialization subsets of spectra in commutative noetherian rings, establishing conditions for abelian categories and cofinite local cohomology modules.

Contribution

It defines ofinitenessor ased on stable under specialization sets and proves abelian category structure and cofinition of local cohomology modules under specific dimensional conditions.

Findings

Category of ofinite modules is abelian in certain cases.

Local cohomology modules are ofinite for specific complexes.

Established conditions for ofiniteness in various dimensional settings.

Abstract

Let $R$ be a commutative noetherian ring, and $\mathcal{Z}$ a stable under specialization subset of $\Spec(R)$. We introduce a notion of $\mathcal{Z}$-cofiniteness and study its main properties. In the case $\dim(\mathcal{Z})\leq 1$, or $\dim(R)\leq 2$, or $R$ is semilocal with $\cd(\mathcal{Z},R) \leq 1$, we show that the category of $\mathcal{Z}$-cofinite $R$-modules is abelian. Also, in each of these cases, we prove that the local cohomology module $H^{i}_{\mathcal{Z}}(X)$ is $\mathcal{Z}$-cofinite for every homologically left-bounded $R$-complex $X$ whose homology modules are finitely generated and every $i \in \mathbb{Z}$.