On the relationship between depth and cohomological dimension

17 Aug 2015  ·  Dao Hailong, Takagi Shunsuke ·

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 $(S_i)$. read more

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Commutative Algebra Algebraic Geometry