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Torus actions on stable module categories, Picard groups, and localizing subcategories | | Akhil Mathew
; | | Date: |
6 Dec 2015 | | Abstract: | 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. | | Source: | arXiv, 1512.1716 | | Services: | Forum | Review | PDF | Favorites | |
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