A family completion theorem for tempered cohomology
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 $I_{\mathbb{G}}(\mathcal{F})=\bigcap_{H\in\mathcal{F}}\mathrm{ker} (\pi_0R({\mathbb{G}})^{ G}\to \pi_0R({\mathbb{G}})^{ H}).$ For ${\mathbb{G}}=\mu_{\mathbb{P}^\infty}$ 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.