Classicality of derived Emerton--Gee stack II: generalised reductive groups
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915531366334464 |
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| author | Min, Yu |
| author_facet | Min, Yu |
| contents | We use the Tannakian formalism to define the Emerton--Gee stack for general groups. For a flat algebraic group G over Z_p, we are able to prove the associated Emerton--Gee stack is a formal algebraic stack locally of finite presentation over Spf(Z_p). We also define a derived stack of Laurent F-crystals with G-structure on the absolute prismatic site, whose underlying classical stack is proved to be equivalent to the Emerton--Gee stack.
In the case of connected reductive groups, we show that the derived stack of Laurent F-crystals with G-structure is classical in the sense that when restricted to truncated animated rings, it is the étale sheafification of the left Kan extension of the Emerton--Gee stack along the inclusion from classical commutative rings to animated rings. Moreover, when G is a generalised reductive group, the classicality result still holds for a modified version of the Emerton--Gee stack. In particular, this completes the picture that the derived stack of local Langlands parameters for the Langlands dual group of a reductive group is classical. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12661 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classicality of derived Emerton--Gee stack II: generalised reductive groups Min, Yu Number Theory Algebraic Geometry We use the Tannakian formalism to define the Emerton--Gee stack for general groups. For a flat algebraic group G over Z_p, we are able to prove the associated Emerton--Gee stack is a formal algebraic stack locally of finite presentation over Spf(Z_p). We also define a derived stack of Laurent F-crystals with G-structure on the absolute prismatic site, whose underlying classical stack is proved to be equivalent to the Emerton--Gee stack. In the case of connected reductive groups, we show that the derived stack of Laurent F-crystals with G-structure is classical in the sense that when restricted to truncated animated rings, it is the étale sheafification of the left Kan extension of the Emerton--Gee stack along the inclusion from classical commutative rings to animated rings. Moreover, when G is a generalised reductive group, the classicality result still holds for a modified version of the Emerton--Gee stack. In particular, this completes the picture that the derived stack of local Langlands parameters for the Langlands dual group of a reductive group is classical. |
| title | Classicality of derived Emerton--Gee stack II: generalised reductive groups |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2411.12661 |