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On the relationship between depth and cohomological dimension

Part of: Local theory Commutative algebra: Homological methods Analytic spaces

Published online by Cambridge University Press:  03 November 2015

Hailong Dao  and
Shunsuke Takagi
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
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Abstract

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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 $(S_{i})$.

Keywords

local cohomologycohomological dimensionlocal Picard groups

MSC classification

Primary: 13D45: Local cohomology 14B15: Local cohomology 32C36: Local cohomology of analytic spaces
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 4 , April 2016 , pp. 876 - 888
Copyright
© The Authors 2015 

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