Torus actions on stable module categories, Picard groups, and localizing subcategories

2015/12/06 by Mathew, Akhil · 1 citation
#Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1512.01716

Abstract

Given an abelian p-group G of rank n, we construct an action of the torus \mathbbTⁿ on the stable module ∞-category of G-representations over a field of characteristic p. The homotopy fixed points are given by the ∞-category of module spectra over the Tate construction of the torus. The relationship thus obtained arises from a Galois extension in the sense of Rognes, with Galois group given by the torus. As one application, we give a homotopy-theoretic proof of Dade's classification of endotrivial modules for abelian p-groups. As another application, we give a slight variant of a key step in the Benson-Iyengar-Krause proof of the classification of localizing subcategories of the stable module category.

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