On the Brauer groups of fibrations II

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 $0\to Sha(Pi{c}_{X/K}^0)\to Br(\mathcal{X})\to Br(X_{\bar{K}}{)}^{G_K}\to 0$, 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.

Merged with arXiv:2103.04945