Classicality of derived Emerton-Gee stack
Yu Min
TL;DR
The paper constructs a derived stack ${\mathcal{X}}$ of Laurent $F$-crystals on the absolute prismatic site and proves its underlying classical stack coincides with the Emerton–Gee stack ${\mathcal{X}}_{EG}$. It then shows ${\mathcal{X}}$ is classical up to nilcompletion by developing a global deformation theory and comparing pro-cotangent spaces via Herr complexes of adjoint étale $(\varphi,\Gamma)$-modules, using Breuil–Kisin prisms and derived representations (Zhu, GV18). This yields ${\mathcal{X}}^{\rm nil} \simeq (\mathrm{Lan}{\mathcal{X}}_{EG})^{\#,\rm nil}$, i.e. the derived extension is governed by the left Kan extension of the classical stack after nilcompletion. The results provide a coherent derived-geometry picture for local Langlands parameters in the ${\mathrm{GL}}_d$-case, suggesting a classical behavior for these derived moduli spaces and furnishing a deformation-theoretic toolkit for future global-to-local comparisons. Overall, the work strengthens the bridge between derived prismatic objects and classical Galois/$(\varphi,\Gamma)$-module moduli, with potential applications to global Langlands correspondences.
Abstract
We construct a derived stack $χ$ of Laurent $F$-crystals on $(\mathcal{O}_K)_{\mathbbΔ}$, 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 $^{\rm cl}χ$ coincides with the Emerton-Gee stack $χ_{\rm EG}$, i.e., the moduli stack of étale $(φ, Γ)$-modules. Then we prove that this derived stack is classical in the sense that when restricted to truncated animated rings, $χ$ is equivalent to the sheafification of the left Kan extension of $χ_{\rm EG}$ along the inclusion from the classical commutative rings to animated rings.