Classicality of derived Emerton-Gee stack

Abstract

We construct a derived stack $\ddot{I}‡$ of Laurent $F$-crystals on $({\mathcal{O}}_K{)}_{\hat{\mathbb{I}}”}$, 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 ${}^{\mathrm{cl}}\ddot{I}‡$ coincides with the Emerton-Gee stack $\ddot{I}{‡}_{\mathrm{EG}}$, i.e., the moduli stack of étale $(\ddot{I}•,\hat{I}“)$-modules. Then we prove that this derived stack is classical in the sense that when restricted to truncated animated rings, $\ddot{I}‡$ is equivalent to the sheafification of the left Kan extension of $\ddot{I}{‡}_{\mathrm{EG}}$ along the inclusion from the classical commutative rings to animated rings.