(Non)Vanishing results on local cohomology of valuation rings

TL;DR

This paper explores local cohomology in valuation rings, establishing vanishing results and highlighting differences between sheaf-theoretic and algebraic definitions in a non-Noetherian context.

Contribution

It provides the first vanishing results for local cohomology in valuation rings of finite Krull dimension, extending the theory beyond Noetherian rings.

Findings

Vanishing of local cohomology in valuation rings of finite Krull dimension.

A uniform bound on the global dimension of such valuation rings.

Differences between sheaf-theoretic and algebraic local cohomology definitions.

Abstract

We examine local cohomology in the setting of valuation rings. The novelty of this investigation stems from the fact that valuation rings are usually non-Noetherian, whereas local cohomology has been extensively developed mostly in a Noetherian setting. We prove various vanishing results on local cohomology for valuation rings of finite Krull dimension. These vanishing results stem from a uniform bound on the global dimension of such rings. Our investigation reveals differences in the sheaf theoretic definition of local cohomology, and the algebraic definition in terms of a limit of certain Ext functors.