On étale hypercohomology of henselian regular local rings with values in $p$-adic étale Tate twists
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| Format: | Preprint |
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2020
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| _version_ | 1866913506696101888 |
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| author | Sakagaito, Makoto |
| author_facet | Sakagaito, Makoto |
| contents | Let $R$ be the henselization of a local ring of a semistable family over the spectrum of a discrete valuation ring of mixed characteristic $(0, p)$ and $k$ the residue field of $R$. In this paper, we prove an isomorphism of étale hypercohomology groups $\operatorname{H}^{n+1}_{\mathrm{\acute{e}t}}(R, \mathfrak{T}_{r}(n)) \simeq \operatorname{H}^{1}_{\mathrm{\acute{e}t}}(k, W_{r}Ω_{\log}^{n})$ for any integers $n\geq 0$ and $r>0$ where $\mathfrak{T}_{r}(n)$ is the $p$-adic Tate twist and $W_{r}Ω_{\log}^{n}$ is the logarithmic Hodge-Witt sheaf. As an application, we prove the local-global principle for Galois cohomology groups over function fields of curves over an excellent henselian discrete valuation ring of mixed characteristic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2002_04797 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On étale hypercohomology of henselian regular local rings with values in $p$-adic étale Tate twists Sakagaito, Makoto Number Theory Let $R$ be the henselization of a local ring of a semistable family over the spectrum of a discrete valuation ring of mixed characteristic $(0, p)$ and $k$ the residue field of $R$. In this paper, we prove an isomorphism of étale hypercohomology groups $\operatorname{H}^{n+1}_{\mathrm{\acute{e}t}}(R, \mathfrak{T}_{r}(n)) \simeq \operatorname{H}^{1}_{\mathrm{\acute{e}t}}(k, W_{r}Ω_{\log}^{n})$ for any integers $n\geq 0$ and $r>0$ where $\mathfrak{T}_{r}(n)$ is the $p$-adic Tate twist and $W_{r}Ω_{\log}^{n}$ is the logarithmic Hodge-Witt sheaf. As an application, we prove the local-global principle for Galois cohomology groups over function fields of curves over an excellent henselian discrete valuation ring of mixed characteristic. |
| title | On étale hypercohomology of henselian regular local rings with values in $p$-adic étale Tate twists |
| topic | Number Theory |
| url | https://arxiv.org/abs/2002.04797 |