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arXiv 2103.04945math.AGmath.NT
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一个应用于Brauer群的$p$-进局部不变循环定理

A $p$-adic local invariant cycle theorem with applications to Brauer groups

Yanshuai Qin

备注：
This preprint has been incorporated into arXiv:2103.06910
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AI总结：

本文证明混合特征下H²的p进局部不变循环定理，推导p进局部域上光滑射影簇的Brauer群相关自然映射的核与余核有限性，将曲线的Brauer群与雅可比簇对偶推广到任意维数。

摘要：

本文证明了混合特征下$H^2$的局部不变循环定理的$p$-进类似结论。由此，对于$p$-进局部域$K$上的光滑射影簇$X$（其在$O_K$上有固有平坦正则模型$\newcommand{\res}{\text{res}} \newcommand{\tr}{\text{tr}} \renewcommand{\theequation}{\fnsymbol{equation}} \newcommand{\bb}[1]{\boldsymbol{#1}} \newcommand{\bZ}{\bb{Z}} \newcommand{\bQ}{\bb{Q}} \newcommand{\bC}{\bb{C}} \newcommand{\bF}{\bb{F}} \newcommand{\fm}{\boldsymbol{\frak{m}}} \newcommand{\fn}{\boldsymbol{\frak{n}}} \newcommand{\fp}{\boldsymbol{\frak{p}}} \newcommand{\bX}{\boldsymbol{\text{X}}} \newcommand{\bY}{\boldsymbol{\text{Y}}} \newcommand{\bG}{\boldsymbol{\text{G}}} \newcommand{\bP}{\boldsymbol{\text{P}}} \newcommand{\bQ}{\boldsymbol{\text{Q}}} \newcommand{\bR}{\boldsymbol{\text{R}}} \newcommand{\bS}{\boldsymbol{\text{S}}} \newcommand{\bT}{\boldsymbol{\text{T}}} \newcommand{\bU}{\boldsymbol{\text{U}}} \newcommand{\bV}{\boldsymbol{\text{V}}} \newcommand{\bW}{\boldsymbol{\text{W}}} \newcommand{\bZ}{\boldsymbol{\text{Z}}} \newcommand{\fa}{\boldsymbol{\frak{a}}} 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英文摘要：

In this article, we prove a $p$-adic analogue of the local invariant cycle theorem for $H^2$ in mixed characteristics. As a result, for a smooth projective variety $X$ over a $p$-adic local field $K$ with a proper flat regular model $\mathcal{X}$ over $O_K$, we show that the natural map $Br(\mathcal{X})\rightarrow Br(X_{\bar{K}})^{G_K}$ has a finite kernel and a finite cokernel. And we prove that the natural map $Hom(Br(X)/Br(K)+Br(\mathcal{X}), \mathbb{Q}/\mathbb{Z}) \rightarrow Alb_X(K)$ has a finite kernel and a finite cokernel, generalizing Lichtenbaum's duality between Brauer groups and Jacobians for curves to arbitrary dimensions.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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