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

Given an abelian $p$-group $G$ of rank $n$, we construct an action of the torus $\mathbb{T}^n$ on the stable module $\infty$-category of $G$-representations over a field of characteristic $p$. The homotopy fixed points are given by the $\infty$-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.