Local cohomological dimension and depth in mixed characteristic

Abstract

Let $(R,\mathfrak m)$ be an unramified regular local ring of mixed characteristic $(0,p)$ and dimension $d$ and let $I\subseteq R$ be an ideal. We prove that $depth(R/I)\geq 3$ implies $cd(I)\leq d-3$, and if $R$ is essentially of finite type over a DVR, then $depth(R/I)\geq 4$ implies $cd(I)\leq d-4$. More generally, $H_I^j(R)$ is a $\mathbb{Q}$-vector space whenever $j>d-depth(R/I)$, thus vanishing of local cohomology in this range is determined completely by the characteristic zero fiber.