A Hodge-Tate decomposition with rigid analytic coefficients
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915721258205184 |
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| author | Gerth, Lucas |
| author_facet | Gerth, Lucas |
| contents | Let $X$ be a smooth proper rigid analytic space over a complete algebraically closed field extension $K$ of $\mathbb{Q}_p$. We establish a Hodge--Tate decomposition for $X$ with $G$-coefficients, where $G$ is any commutative locally $p$-divisible rigid group. This generalizes the Hodge--Tate decomposition of Faltings and Scholze, which is the case $G=\mathbb{G}_a$. For this, we introduce geometric analogs of the Hodge--Tate spectral sequence with general locally $p$-divisible coefficients. We prove that these spectral sequences degenerate at $E_2$. Our results apply more generally to a class of smooth families of commutative adic groups over $X$ and in the relative setting of smooth proper morphisms $X\rightarrow S$ of smooth rigid spaces. We deduce applications to analytic Brauer groups and the geometric $p$-adic Simpson correspondence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07366 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Hodge-Tate decomposition with rigid analytic coefficients Gerth, Lucas Algebraic Geometry Let $X$ be a smooth proper rigid analytic space over a complete algebraically closed field extension $K$ of $\mathbb{Q}_p$. We establish a Hodge--Tate decomposition for $X$ with $G$-coefficients, where $G$ is any commutative locally $p$-divisible rigid group. This generalizes the Hodge--Tate decomposition of Faltings and Scholze, which is the case $G=\mathbb{G}_a$. For this, we introduce geometric analogs of the Hodge--Tate spectral sequence with general locally $p$-divisible coefficients. We prove that these spectral sequences degenerate at $E_2$. Our results apply more generally to a class of smooth families of commutative adic groups over $X$ and in the relative setting of smooth proper morphisms $X\rightarrow S$ of smooth rigid spaces. We deduce applications to analytic Brauer groups and the geometric $p$-adic Simpson correspondence. |
| title | A Hodge-Tate decomposition with rigid analytic coefficients |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2411.07366 |