Cohomological methods in integral representation theory
type: Item
instance of: scholarly article
title: Cohomological methods in integral representation theory
zbMATH Open document ID: 0691.20009
DOI: 10.1016/0022-4049(89)90142-4
author: Peter Symonds
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let R be a Dedekind domain of characteristic zero and let G be a finite group. A cohomological criterion for an RG-lattice M to have a nonzero sublattice of the trivial lattice as a direct summand is given; namely \(\hat H^ 0(G;M)\) must contain a copy of R/\(| G| R\). It is shown that there is an isomorphism \(\tilde H^ i(G;M^*)\cong \hat H^{- i}(G;M)\) in Tate cohomology, induced by the cup-product. There follows a discussion of cohomological features of the stable lattice category of RG; criteria for objects to be direct summands of syzygies in that category are given. A cohomological criterion for isomorphism in the stable category follows as a corollary. Motivated by topological constructions, the author shows how to construct new lattices called the path space and the cone space from any homomorphism of lattices. Finally, the ideas are used to discuss a cobordism invariant for certain manifolds with group actions.
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 55S37
zbMATH DE Number: 4130643
zbMATH Keywords: finite group
zbMATH Keywords: RG-lattice
zbMATH Keywords: Tate cohomology
zbMATH Keywords: cup-product
zbMATH Keywords: stable lattice category
zbMATH Keywords: direct summands of syzygies
zbMATH Keywords: manifolds with group actions
reviewed by: William H. Gustafson
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90142-4
OpenAlex ID: W1992567761
cites work: Q4143495
cites work: Q3922829
cites work: Homotopical algebra
cites work: Periodic resolutions for finite groups
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description: scientific article; zbMATH DE number 4130643