2007/04/16 by Peter Schenzel, Schenzel, Peter
Mathematics · #13D45 #14B15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #math.AC #math.AG #msc:13D45 #msc:14B15
paper · pdf · doi:10.48550/arxiv.0704.2005
26 pages
arxiv created 2007/04/16 · openalex publication_date 2007/04/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \HHi(M/\mathfrak anM), i ∈ \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 ∖ 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.