A family completion theorem for tempered cohomology

2026/08/03 by Leonard Tokic
Mathematics · #math.AT

paper · pdf

44 pages, comments welcome!

arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

Let \mathbbG be an oriented ℙ-divisible group over a noetherian 𝔼_∞-ring R, let G be a finite group, and let F be a family of subgroups of G. We show that completion of R(\mathbbG)_G-modules at F agrees with algebraic completion at the ideal I_\mathbbG(F)=\bigcap_H\inFker (π₀R(\mathbbG) ^G→ π₀R(\mathbbG) ^H). For \mathbbG=μ_ℙ^∞ over 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 \mathbbG\\mathbbB 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 F are exactly the preimage of V(I_\mathbbG(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 \mathbbG(\mathbbB G), the analogue over such a base of Spec R(\mathbbG) ^G, and which applies for instance to genuine equivariant topological modular forms.

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