Algebraic invariants for finite group actions. I: Varieties
The paper concerns applications of shifted subgroups, cohomological support varieties, rank varieties and complexity to finite transformation groups. The author studies the cohomology of a finite group with coefficients in a cochain complex (see, e.g., \textit{K. S. Brown}, Cohomology of groups (1982; Zbl 0584.20036), Ch. VII, Sec. 5) with emphasis on elementary abelian \(p\)-groups and cochain complexes with finitely generated cohomology. He proves an algebraic version of the Localization Theorem of Borel, Hsiang, Quillen, which can be applied topologically to elementary abelian \(p\)-group actions on spaces with finite dimensional mod. \(p\) cohomology. Among the various applications there are (1) a Rim-Chouinard-Dade type projectivity criterion for a \(G\)- module arising as the reduced homology of a \(G\)-space; (2) necessary and sufficient conditions for certain CW-complexes to be finitely dominated; and (3) some Borsuk-Ulam type theorems.
elementary abelian \(p\)-group actions on spaces with finite dimensional mod. \(p\) cohomology
cohomology of a finite group with coefficients in a cochain complex
scientific article; zbMATH DE number 490808