Modeling Group Actions on Stacks (Especially the Lubin-Tate Action)

Abstract

Suppose we are given a profinite group G acting on a formal moduli stack M, and we want to understand the group action, and compute cohomology related to this group action. How can we do it? This prolegomenon conveys two methods of pinning down such an action: geometric modeling and the two-tower method. We highlight their use on a specific action -- the automorphisms of a formal group acting on its deformation space, called the Lubin-Tate action. We provide the toolkit that these two methods run on. The pieces of the kit are standard in their own subjects, but the Lubin-Tate action requires us to sew them together and ask more of them than their usual statements give, and several are set down here in that stronger form: equivariantly, in families, and with the grading kept. The tools are put to work in the two sequels, which build geometric models at height p-1 and at the heights p^{k-1}(p-1) above it; the two-tower material is a companion to the proof of the coheight one weak chromatic splitting conjecture.

arXiv commentAbstract rewritten; body broken into three parts and substantially expanded: general material formerly carried in arXiv:2509.23428 is collected here as toolkits, stated once in the generality the later papers use; new no-go obstruction, Lubin-Tate tangent as a G'-module at height p^{k-1}(p-1), and ideal-hop lemma; profinite foundations corrected to align with the smooth/condensed comparison