Top local cohomology and the catenaricity of the unmixed support of a finitely generated module

Details Top local cohomology and the catenaricity of the unmixed support of a finitely generated module Top local cohomology and the catenaricity of the unmixed support of a finitely generated module

2007-01-24

arxiv.org/abs/math/0701675v1

Mathematics

Commutative Algebra

11 pages

Scientific paper

Let $(R,\m)$ be a Noetherian local ring and $M$ a finitely generated $R-$module with $\dim M=d.$ This paper is concerned with the following property for the top local cohomology module $H^d_\m(M)$: $$\Ann (0:_{H^d_\m(M)}\p)=\p\ \text{for all prime ideals} \p\supseteq\Ann H^d_\m(M).$$ In this paper we will show that this property is equivalent to the catenaricity of the unmixed support $\Usupp M$ of $M$ which is defined by $\Usupp M=\Supp M/U_M(0)$, where $U_M(0)$ is the largest submodule of $M$ of dimension less than $d.$ Some characterizations of this property in terms of system of parameters as well as the relation between the unmixed supports of $M$ and of the $\m$-adic completion $\hat M$ are given.

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