On the relationship between depth and cohomological dimension

Let $(S,\mathfrak{m})$ be an $n$ -dimensional regular local ring essentially of finite type over a field and let $\mathfrak{a}$ be an ideal of $S$ . We prove that if $\text{depth}\,S/\mathfrak{a}\geqslant 3$ , then the cohomological dimension $\text{cd}(S,\mathfrak{a})$ of $\mathfrak{a}$ is less than or equal to $n-3$ . This settles a conjecture of Varbaro for such an $S$ . We also show, under the assumption that $S$ has an algebraically closed residue field of characteristic zero, that if $\text{depth}\,S/\mathfrak{a}\geqslant 4$ , then $\text{cd}(S,\mathfrak{a})\leqslant n-4$ if and only if the local Picard group of the completion $\widehat{S/\mathfrak{a}}$ is torsion. We give a number of applications, including a vanishing result on Lyubeznik’s numbers, and sharp bounds on the cohomological dimension of ideals whose quotients satisfy good depth conditions such as Serre’s conditions $(S_{i})$ .