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On the relationship between depth and cohomological dimension

Published online by Cambridge University Press:  03 November 2015

Hailong Dao
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS 66045-7523, USA email hdao@ku.edu
Shunsuke Takagi
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan email stakagi@ms.u-tokyo.ac.jp
Corresponding

Abstract

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 $(S_{i})$ .

Type
Research Article
Copyright
© The Authors 2015 

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