Vanishing of local cohomology in unramified mixed characteristic

TL;DR

This paper extends Faltings' bounds on the cohomological dimension of ideals from equal characteristic to unramified mixed characteristic, providing sharp bounds in this broader setting.

Contribution

It generalizes Faltings' bound on cohomological dimension to unramified mixed characteristic regular local rings, establishing sharpness of the bound.

Findings

Extended Faltings' bound to mixed characteristic

Proved the bound is sharp in this setting

Established a link between cohomological dimension and topological invariants

Abstract

Given an ideal $I$ in a regular local ring $A$, the cohomological dimension of $I$ in $A$ is the index of the highest non-vanishing local cohomology of $A$ supported at $I$. Determining effective upper bounds on the cohomological dimension in terms of topological invariants of $\text{Spec}(A/I)$ is a central problem in commutative algebra. In equal characteristic, Faltings proved in 1980 a general bound on the cohomological dimension of an ideal in terms of its big height. In this article, we extend Faltings' result to the unramified mixed characteristic setting and show that the resulting bound is sharp.