Cohomological methods in integral representation theory
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.
scientific article; zbMATH DE number 4130643
scientific article