A family completion theorem for tempered cohomology

Let $\mathbb{G}$ be an oriented $\mathbb{P}$-divisible group over a noetherian ${\mathbb{E}}_{\mathrm{\infty }}$-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 $I_{\mathbb{G}}(\mathcal{F})={\bigcap }_{H\in \mathcal{F}}\mathrm{k}\mathrm{e}\mathrm{r}({\pi }_{0}R(\mathbb{G}{)}^G\to {\pi }_{0}R(\mathbb{G}{)}^H)\mathrm{.}$ For $\mathbb{G}={\mu }_{{\mathbb{P}}^{\mathrm{\infty }}}$ over $\mathrm{K}\mathrm{U}$ 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 $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R(\mathbb{G}{)}^G$, and which applies for instance to genuine equivariant topological modular forms.