A Nomizu-van Est theorem in Ekedahl's derived $\ell$-adic setting
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866909986849816576 |
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| author | Taïbi, Olivier |
| author_facet | Taïbi, Olivier |
| contents | A theorem of Nomizu and van Est computes the cohomology of a compact nilmanifold, or equivalently the group cohomology of an arithmetic subgroup of a unipotent linear algebraic group over $\mathbb{Q}$. We prove a similar result for the cohomology of a compact open subgroup of a unipotent linear algebraic group over $\mathbb{Q}_{\ell}$ with coefficients in a complex of continuous $\ell$-adic representations. We work with the triangulated categories defined by Ekedahl which play the role of ``derived categories of continuous $\ell$-adic representations''. This is motivated by Pink's formula computing the derived direct image of an $\ell$-adic local system on a Shimura variety in its minimal compactification, and its application to automorphic perverse sheaves on Shimura varieties. The key technical result is the computation of the cohomology with coefficients in a unipotent representation with torsion coefficients by an explicit complex of polynomial cochains which is of finite type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_09744 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Nomizu-van Est theorem in Ekedahl's derived $\ell$-adic setting Taïbi, Olivier Number Theory Algebraic Geometry A theorem of Nomizu and van Est computes the cohomology of a compact nilmanifold, or equivalently the group cohomology of an arithmetic subgroup of a unipotent linear algebraic group over $\mathbb{Q}$. We prove a similar result for the cohomology of a compact open subgroup of a unipotent linear algebraic group over $\mathbb{Q}_{\ell}$ with coefficients in a complex of continuous $\ell$-adic representations. We work with the triangulated categories defined by Ekedahl which play the role of ``derived categories of continuous $\ell$-adic representations''. This is motivated by Pink's formula computing the derived direct image of an $\ell$-adic local system on a Shimura variety in its minimal compactification, and its application to automorphic perverse sheaves on Shimura varieties. The key technical result is the computation of the cohomology with coefficients in a unipotent representation with torsion coefficients by an explicit complex of polynomial cochains which is of finite type. |
| title | A Nomizu-van Est theorem in Ekedahl's derived $\ell$-adic setting |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2209.09744 |