On the relationship between depth and cohomological dimension

TL;DR

This paper establishes bounds on the cohomological dimension of ideals in regular local rings based on the depth of their quotients, linking algebraic properties to geometric invariants like the Picard group.

Contribution

It proves new bounds relating depth and cohomological dimension, and characterizes when these bounds are sharp using the Picard group in characteristic zero.

Findings

If depth S/I ≥ 3, then cd(S, I) ≤ n-3.

If depth S/I ≥ 4 and residue field is algebraically closed of characteristic zero, then cd(S, I) ≤ n-4 iff the local Picard group is torsion.

Provides applications with sharp bounds on cohomological dimension for ideals satisfying Serre's conditions.

Abstract

Let $(S, m)$ be an $n$-dimensional regular local ring essentially of finite type over a field and let $I$ be an ideal of $S$. We prove that if $\text{depth} S/I \ge 3$, then the cohomological dimension $\mathrm{cd}(S, I)$ of $I$ is less than or equal to $n-3$. We also show, under the assumption that $S$ has an algebraically closed residue field of characteristic zero, that if $\text{depth} S/I \ge 4$, then $\mathrm{cd}(S, I) \le n-4$ if and only if the local Picard group of the completion $\widehat{S/I}$ is torsion. We give a number of applications, including sharp bounds on cohomological dimension of ideals whose quotients satisfy good depth conditions such as Serre's conditions $(S_i)$.