2021/05/06 by David Ayala, Ayala, David, Aaron Mazel-Gee +3 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2105.02456
openalex publication_date 2021/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish a novel approach to computing G-equivariant cohomology for a finite group G, and demonstrate it in the case that G = Cpn. For any commutative ring spectrum R, we prove a symmetric monoidal reconstruction theorem for genuine G-R-modules, which records them in terms of their geometric fixedpoints as well as gluing maps involving their Tate cohomologies. This reconstruction theorem follows from a symmetric monoidal stratification (in the sense of \citeAMR-strat); here we identify the gluing functors of this stratification in terms of Tate cohomology. Passing from genuine G-spectra to genuine G-ℤ-modules (a.k.a. derived Mackey functors) provides a convenient intermediate category for calculating equivariant cohomology. Indeed, as ℤ-linear Tate cohomology is far simpler than \mathbbS-linear Tate cohomology, the above reconstruction theorem gives a particularly simple algebraic description of genuine G-ℤ-modules. We apply this in the case that G = Cpn for an odd prime p, computing the Picard group of genuine G-ℤ-modules (and therefore that of genuine G-spectra) as well as the RO(G)-graded and Picard-graded G-equivariant cohomology of a point.