On the Brauer groups of fibrations II

TL;DR

This paper generalizes a classical theorem relating the Brauer group of a scheme over a number field to its geometric and arithmetic properties, extending results from surfaces to higher dimensions.

Contribution

It establishes an exact sequence connecting the Brauer group of a proper regular flat scheme over a number field to its geometric Brauer group, generalizing Artin-Grothendieck's theorem to arbitrary dimensions.

Findings

Established an exact sequence relating $Br(\\mathcal{X})$, $Br(X_{\bar{K}})$, and $Sha(Pic_{X/K}^0)$.

Reduced the problem of finiteness of $Br(\ ext{scheme})$ over $\\mathbb{Z}$ to the case of 3-dimensional schemes.

Extended classical results from arithmetic surfaces to higher-dimensional schemes.

Abstract

Let $K$ be a number field, and let $\mathcal{X}$ be a proper regular flat scheme over $\mathcal{O}_{K}$ with a generic fiber $X$ geometrically connected over $K$. We prove that there is an exact sequence up to finite groups $0\rightarrow Sha(Pic_{X/K}^0)\rightarrow Br(\mathcal{X})\rightarrow Br(X_{\bar{K}})^{G_K}\rightarrow 0$, which generalizes a theorem of Artin and Grothendieck for arithmetic surfaces to arbitrary dimensions. Consequently, we reduce Artin's question regarding the finiteness of $Br(\mathcal{X})$ for proper regular flat schemes $\mathcal{X}$ over $\mathbb{Z}$ to $3$-dimensional arithmetic schemes.