关于以$p$-adic étale Tate twist为值的henselian正则局部环的étale上同调
On étale hypercohomology of henselian regular local rings with values in $p$-adic étale Tate twists
AI总结：
本文证明henselian正则局部环的étale上同调与剩余域上对数Hodge-Witt层的同构，并应用于混合特征曲线函数域的Galois上同调局部-全局原理。
摘要：
设$R$为混合特征$(0,p)$离散赋值环谱上半稳定族局部环的hensel化，$k$为$R$的剩余域。本文证明了对于任意整数$n\geq 0$和$r>0$，étale上同调群之间存在同构$\operatorname{H}^{n+1}_{\mathrm{\acute{e}t}}(R, \mathfrak{T}_{r}(n)) \simeq \operatorname{H}^{1}_{\mathrm{\acute{e}t}}(k, W_{r}Ω_{\log}^{n})$，其中$\mathfrak{T}_{r}(n)$为$p$-adic Tate twist，$W_{r}Ω_{\log}^{n}$为对数Hodge-Witt层。作为应用，我们证明了混合特征优良henselian离散赋值环上曲线函数域的Galois上同调群的局部-全局原理。
英文摘要：
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.
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