On the relationship between depth and cohomological dimension

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 $(S_i)$.

The Proposition 2.3 of our previous version was not correct as stated. We thank Bhargav Bhatt for pointing this out. It has been fixed in this version. Some minor changes were also added to improve clarity

References in corpus (2)

• Cohomological and projective dimensions
• Grothendieck--Lefschetz type theorems for the local Picard group

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