AI 中文总结
本文在导出代数几何中定义参数化P-可除群常有限阿贝尔子群的商栈，统一推广拓扑K-理论与拓扑模形式上的伽罗瓦作用，得到K-几何不动点的伽罗瓦扩张，并将等变环谱的完美模∞-范畴分解，得到其局部化不变量的Mayer-Vietoris序列。
AI 中文摘要
本文统一并推广了与本原n次单位根相伴的拓扑K-理论上的分圆伽罗瓦作用，以及具有Γ₁(n)和Γ(n)级结构的拓扑模形式上著名的GL₁(Z/n)和GL₂(Z/n)伽罗瓦作用。这通过在导出代数几何中定义一个参数化P-可除群的常有限阿贝尔子群的商栈，并研究这些商栈与各类挠子概念如何通过取代数和范畴不变量形成的伽罗瓦扩张相互作用来实现。取整体截面后得到标题所述的K-几何不动点上的伽罗瓦扩张，恢复并改进了上述熟知例子，还提供了新的例子。作为应用，多种H-等变环谱R的完美模的∞-范畴被分解为非等变范畴的简单拉回，得到R的局部化不变量的Mayer-Vietoris序列。例如，这对所有p-群以及阶小于500的任何有限非阿贝尔单群的等变拓扑K-理论均成立。
英文摘要
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 $Γ_1(n)$- and $Γ(n)$-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.
Comments55 pages. Comments always welcome
