2021/02/04 by Eric M. Friedlander, Friedlander, Eric M.
Mathematics · #16G99 #16S40 #16T05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2102.02453
openalex publication_date 2021/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Following earlier work with Cris Negron on the cohomology of Drinfeld doubles D(\mathbb G(r)), we develop a "geometric theory" of support varieties for "extended Drinfeld doubles" D(\mathbb G(r)) of Frobenius kernels \mathbb G(r) of smooth linear algebraic groups \mathbb G over a field k of characteristic p > 0. To a D(\mathbb G(r))-module M we associate the space Π( D(\mathbb G(r)))M of equivalence classes of "pairs of π-points" and prove most of the desired properties of M ↦ Π( D(\mathbb G(r)))M. Namely, this association satisfies the "tensor product property" and admits a natural continuous map Ψ D to cohomological support theory. Moreover, for M finite dimensional and with suitable conditions on \mathbb G(r), this association provides a "projectivity test", Ψ D is a homeomorphism, and identifies Π( D(\mathbb G(r)))M with the cohomological support variety of M for various classes of D(\mathbb G(r))-modules M.