  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.
