Classicality of derived Emerton--Gee stack II: generalised reductive groups

TL;DR

This paper extends the Emerton--Gee stack to general groups using Tannakian formalism, proving its algebraic structure and classicality in the derived setting, especially for reductive and generalised reductive groups.

Contribution

It defines the Emerton--Gee stack for general groups via Tannakian formalism and proves its classicality for reductive and generalised reductive groups.

Findings

The Emerton--Gee stack is a formal algebraic stack over Z_p.

The derived stack of Laurent F-crystals with G-structure is classical for reductive groups.

Classicality holds for a modified stack in the case of generalised reductive groups.

Abstract

We use the Tannakian formalism to define the Emerton--Gee stack for general groups. For a flat algebraic group G over Z_p, we are able to prove the associated Emerton--Gee stack is a formal algebraic stack locally of finite presentation over Spf(Z_p). We also define a derived stack of Laurent F-crystals with G-structure on the absolute prismatic site, whose underlying classical stack is proved to be equivalent to the Emerton--Gee stack. In the case of connected reductive groups, we show that the derived stack of Laurent F-crystals with G-structure is classical in the sense that when restricted to truncated animated rings, it is the \'etale sheafification of the left Kan extension of the Emerton--Gee stack along the inclusion from classical commutative rings to animated rings. Moreover, when G is a generalised reductive group, the classicality result still holds for a modified version of the Emerton--Gee stack. In particular, this completes the picture that the derived stack of local Langlands parameters for the Langlands dual group of a reductive group is classical.