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arXiv · 2309.05066

Classicality of derived Emerton-Gee stack

Yu Min
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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.

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Yu Min. 2023-09-10. Classicality of derived Emerton-Gee stack. https://arxiv.org/abs/2309.05066

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