Moduli stacks of Lubin--Tate $(\varphi,\Gamma)$-modules

TL;DR

This paper introduces stacks parametrizing Lubin--Tate $(, abla)$-modules, compares them with cyclotomic cases at perfectoid level, and proves the Herr complex's perfectness in this context.

Contribution

It defines and studies Lubin--Tate $(, abla)$-module stacks and relates them to Emerton--Gee stacks, establishing new connections and properties.

Findings

Comparison between Lubin--Tate and cyclotomic stacks at perfectoid level

Proof of perfectness of the Herr complex in the Lubin--Tate setting

Establishment of a moduli stack framework for Lubin--Tate $(, abla)$-modules

Abstract

We define and study stacks which parametrize Lubin--Tate $(\varphi,\Gamma)$-modules. By working at a perfectoid level, we compare these with the Emerton--Gee stacks of cyclotomic $(\varphi,\Gamma)$-modules. As a consequence, we deduce perfectness of the Herr complex in the Lubin--Tate setting.