Evaluating the tame Brauer group of open varieties over local fields
Victor de Vries
TL;DR
The paper analyzes evaluation maps on the p-torsion and p'-torsion parts of the Brauer group for open varieties over local fields, where the complement is a divisor. It shows that Leray-spectral-sequence-based arguments are not in general sufficient to determine the evaluation maps, and develops a derived-category framework using localisation triangles, Gysin maps, and representable functors to articulate how $u^*$ is controlled by geometric data. The main result states that $u^*$ depends only on the cohomology class $\mathrm{cl}(Z\cap x)\in \mathrm{H}^{2d}_{|Z\cap x|}(Z,\mu_n^{\otimes d})$, effectively reducing the problem to local intersection data; the argument proceeds by comparing two localisation triangles and extracting the key component $\alpha$, which is shown to be determined by cycle-classes via base-change compatibility. The paper also discusses special cases (e.g. snc divisors with $\gcd(n,q-1)=1$) where the analysis simplifies, and highlights open questions about the sharpness of the conditions and possible extensions.
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 $u^*:\mathrm{H}^2(U,μ_n)\to \mathrm{H}^2(k_v,μ_n)$. 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 $\mathrm{H}^{2d}_{|\overline{x}|}(Z,μ_n^{\otimes d})$.