On Galois extensions of geometric fixed point spectra

Abstract

In this article, the cyclotomic Galois action on topological K-theory adjoined with a primitive $n$th root of unity and the famous $GL_1(\mathbf{Z}/n)$- and $GL_2(\mathbf{Z}/n)$-Galois actions on topological modular forms with $\Gamma_1(n)$- and $\Gamma(n)$-level structures are unified and generalised. This is done by defining a quotient stack in derived algebraic geometry parametrising constant finite abelian subgroups of $\mathbf{P}$-divisible groups, and studying how these quotient stacks and various notions of torsors interact with Galois extensions formed by taking algebraic and categorical invariants. Taking global sections then yields the titular Galois extensions on $K$-geometric fixed points, recovering and refining the well-known examples above and providing new ones. As an application, the $\infty$-category of perfect modules over a variety of $H$-equivariant ring spectra $R$ are decomposed into simple pullbacks of nonequivariant categories, leading to Mayer--Vietoris sequences for localising invariants of $R$. For example, this occurs for equivariant topological K-theory for all $p$-groups as well as any finite nonabelian simple group of order less than 500.