Evaluating the tame Brauer group of open varieties over local fields

TL;DR

This paper investigates the evaluation map of the Brauer group on open varieties over local fields, establishing conditions under which evaluations at different points coincide, with implications for understanding the tame Brauer group.

Contribution

It provides new criteria for when evaluation maps of the Brauer group are equal for points lifting to the same reduction and intersection class in a regular scheme over local fields.

Findings

Evaluation maps coincide under specific reduction and intersection conditions.

Conditions relate to the reduction of points and their intersection classes in the divisor.

Results enhance understanding of the tame Brauer group over local fields.

Abstract

In this document we let $U$ be a smooth variety of pure dimension $d$ over a local field $k_v$ with unit ball $\mathcal{O}_v$ and residue field $\mathbb{F}$ of characteristic $p>0$ and we set $n$ to be a positive integer such that $p\nmid n$. For various $u\in U(k_v)$ we study the evaluation map $u^*:\mathrm{H}^2(U,\mu_n)\to \mathrm{H}^2(k_v,\mu_n)$. We suppose that $U$ embeds as an open subscheme in a regular scheme $\mathcal{X}$ that is of finite type over $\mathcal{O}_v$. We assume that $Z:=\mathcal{X}\setminus U$ is a divisor and we endow it with its reduced scheme structure. We show that for $u_1,u_2\in U(k_v)$ that lift to $x_1,x_2\in \mathcal{X}(\mathcal{O}_v)$ we obtain the same evaluation map $u_1^*=u_2^*$ under the two conditions that first, there is an equality of reductions $\overline{x_1}=\overline{x_2}$ in $\mathcal{X}(\mathbb{F})$ and second, that $\mathrm{cl}(x_1\cap Z)=\mathrm{cl}(x_2\cap Z)$ holds in $\mathrm{H}^{2d}_{|\overline{x}|}(Z,\mu_n^{\otimes d})$.