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 $0\rightarrow Sha(Pic_{X/K}^0)\rightarrow Br(\mathcal{X})\rightarrow Br(X_{\bar{K}})^{G_K}\rightarrow 0$, 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.