Classicality of derived Emerton-Gee stack

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.