Classicality of derived Emerton-Gee stack
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866915280611966976 |
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| author | Min, Yu |
| author_facet | Min, Yu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_05066 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classicality of derived Emerton-Gee stack Min, Yu Number Theory 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. |
| title | Classicality of derived Emerton-Gee stack |
| topic | Number Theory |
| url | https://arxiv.org/abs/2309.05066 |