On the Brauer groups of fibrations II

2021/03/11 by Qin, Yanshuai
#14F22 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2103.06910

Abstract

Let K be a number field, and let X be a proper regular flat scheme over O_K with a generic fiber X geometrically connected over K. We prove that there is an exact sequence up to finite groups 0→ Sha(Pic_X/K⁰)→ Br(X)→ Br(X_K)^G_K→ 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(X) for proper regular flat schemes X over ℤ to 3-dimensional arithmetic schemes.

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