Classicality of derived Emerton-Gee stack

TL;DR

This paper constructs a derived stack of Laurent F-crystals over the Emerton-Gee stack and proves its classicality, linking derived and classical moduli stacks of Galois representations.

Contribution

It introduces a derived stack of Laurent F-crystals and demonstrates its equivalence to the classical Emerton-Gee stack, establishing its classical nature in derived algebraic geometry.

Findings

The derived stack $oldsymbol{ ext{χ}}$ coincides with the classical Emerton-Gee stack.

The derived stack $oldsymbol{ ext{χ}}$ is classical when restricted to truncated animated rings.

The construction bridges derived and classical moduli of Galois representations.

Abstract

We construct a derived stack $\chi$ of Laurent $F$-crystals on $(\mathcal{O}_K)_{\mathbb{\Delta}}$, where $\mathcal{O}_K$ is the ring of integers of a finite extension $K$ of $\mathcal{Q}_p$. We first show that its underlying classical stack $^{\rm cl}\chi$ coincides with the Emerton-Gee stack $\chi_{\rm EG}$, i.e., the moduli stack of \'etale $(\phi, \Gamma)$-modules. Then we prove that this derived stack is classical in the sense that when restricted to truncated animated rings, $\chi$ is equivalent to the sheafification of the left Kan extension of $\chi_{\rm EG}$ along the inclusion from the classical commutative rings to animated rings.