On the formal cohomology of local rings

Peter Schenzel

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arXiv · 0704.2005

On the formal cohomology of local rings

Peter Schenzel

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Abstract

Let $\mathfrak a$ denote an ideal of a local ring $(R, \mathfrak m).$ Let $M$ be a finitely generated $R$-module. There is a systematic study of the formal cohomology modules $\varprojlim \HH^i(M/\mathfrak a^nM), i \in \mathbb Z.$ We analyze their $R$-module structure, the upper and lower vanishing and non-vanishing in terms of intrinsic data of $M,$ and its functorial behavior. These cohomology modules occur in relation to the formal completion of the punctured spectrum $\Spec R \setminus V(\mathfrak m).$ As a new cohomological data there is a description on the formal grade $\fgrade(\mathfrak a, M)$ defined as the minimal non-vanishing of the formal cohomology modules. There are various exact sequences concerning the formal cohomology modules. Among them a Mayer-Vietoris sequence for two ideals. It applies to new connectedness results. There are also relations to local cohomological dimensions.

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Peter Schenzel. 2007-04-16. On the formal cohomology of local rings. https://arxiv.org/abs/0704.2005

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