关于 tempered 上同调的族完备化定理
A family completion theorem for tempered cohomology
浏览论文内容
中文总结 AI 辅助
本文在诺特$\mathbb{E}_\infty$-环上的定向$\mathbb{P}$-可除群框架下，证明tempered上同调的族完备化定理，推广了经典等变上同调完备化结果，并拓展至局部诺特几何基栈情形。
中文摘要 AI 辅助
设${\mathbb{G}}$是诺特$\mathbb{E}_\infty$-环$R$上的一个定向$\mathbb{P}$-可除群，设$G$是一个有限群，$\mathcal{F}$是$G$的一个子群族。我们证明，$R({\mathbb{G}})_G$-模在$\mathcal{F}$处的完备化与在理想$I_{\mathbb{G}}(\mathcal{F})=\bigcap_{H\in\mathcal{F}}\mathrm{ker} (π_0R({\mathbb{G}})^{ G}\to π_0R({\mathbb{G}})^{ H})$处的代数完备化一致。当${\mathbb{G}}$为$\mathrm{KU}$上的$μ_{\mathbb{P}^\infty}$时，这一结果恢复了Adams、Haeberly、Jackowski和May的族完备化定理；当子群族为平凡族时，则恢复了经典的Atiyah-Segal完备化定理。研究的核心输入是tempered特征栈${\mathbb{G}}\{\mathbb{B} G\}$的点的支集理论，其思路源自Segal对复表示环素谱的分析：我们证明一个点的支集是$G$的阿贝尔子群的单个共轭类，且支集包含于$\mathcal{F}$的点恰好是仿射化映射下$V(I_{\mathbb{G}}(\mathcal{F}))$的原像。我们还证明了局部诺特几何基栈上的一个版本，其中理想被替换为${\mathbb{G}}({\mathbb{B}} G)$的一个开子栈——它是$\mathrm{Spec}\\,R({\mathbb{G}})^{ G}$在此类基上的类似物——该版本可应用于例如真正等变拓扑模形式。
英文摘要
Let ${\mathbb{G}}$ be an oriented $\mathbb{P}$-divisible group over a noetherian $\mathbb{E}_\infty$-ring $R$, let $G$ be a finite group, and let $\mathcal{F}$ be a family of subgroups of $G$. We show that completion of $R({\mathbb{G}})_G$-modules at $\mathcal{F}$ agrees with algebraic completion at the ideal $I_{\mathbb{G}}(\mathcal{F})=\bigcap_{H\in\mathcal{F}}\mathrm{ker} (π_0R({\mathbb{G}})^{ G}\to π_0R({\mathbb{G}})^{ H}).$ For ${\mathbb{G}}=μ_{\mathbb{P}^\infty}$ over $\mathrm{KU}$ this recovers the family completion theorem of Adams, Haeberly, Jackowski, and May, and for the trivial family the classical Atiyah-Segal completion theorem. The main input is a theory of support for points of the tempered character stack ${\mathbb{G}}\{\mathbb{B} G\}$, in the spirit of Segal's analysis of the prime spectrum of the complex representation ring: we show that the support of a point is a single conjugacy class of abelian subgroups of $G$, and that the points supported inside $\mathcal{F}$ are exactly the preimage of $V(I_{\mathbb{G}}(\mathcal{F}))$ under the affinization map. We also prove a version over locally noetherian geometric base stacks, in which the ideal is replaced by an open substack of ${\mathbb{G}}({\mathbb{B}} G)$, the analogue over such a base of $\mathrm{Spec}\,R({\mathbb{G}})^{ G}$, and which applies for instance to genuine equivariant topological modular forms.
