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

TL;DR

This paper explores methods to analyze profinite group actions on formal moduli stacks, focusing on the Lubin-Tate action, and discusses geometric modeling and the two tower method for understanding such actions and their cohomology.

Contribution

It introduces and compares two approaches—geometric modeling and the two tower method—for understanding group actions on stacks, with detailed application to Lubin-Tate deformation spaces.

Findings

Demonstrates the application of geometric modeling to Lubin-Tate actions.

Shows how the two tower method can be used to compute cohomology of group actions.

Provides insights into the structure of automorphism groups acting on formal moduli stacks.

Abstract

Suppose we are given a profinite group $G$ acting on a formal moduli stack $\mathcal{M}$, and we want to understand the group action, and compute cohomology related to this group action. How can we do it? This prolegomenon surveys 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.