摘要翻译：
设$\mathfrak a$表示局部环$(R，\mathfrak m)的理想。$Let$m$是有限生成的$R$-模块。本文系统地研究了形式上同调模$\varprojlim\hh^i(M/\mathfrak a^nm),i\in\mathbb z。我们从$m,$的内在数据及其函数行为出发，分析了它们的$r$-模结构、上下消失和非消失。这些上同调模与删余谱的形式完备有关。作为一种新的上同调数据，定义了形式上同调模的最小不消失的形式等级$\fgrade(\mathfrak a,m）。关于形式上同调模有各种精确序列。其中，梅耶尔-越多的两个理想序列。它适用于新的连通性结果。与局部上同调维数也有关系。
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英文标题：
《On the formal cohomology of local rings》
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作者：
Peter Schenzel
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Commutative Algebra        交换代数
分类描述：Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环，模，理想，同调代数，计算方面，不变理论，与代数几何和组合学的联系
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  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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PDF链接：
https://arxiv.org/pdf/0704.2005