Vanishing of local cohomology in unramified mixed characteristic

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.