The algebraic Brauer group of a pinched variety

2022/09/14 by Gonzalez-Aviles, Cristian D.
#11G20 (Secondary) #14B05 #14F22 (Primary) #14H20 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2209.06653

Abstract

Added lemma provided by Michel Brion. Other (minor) changes. Submitted version. Let k be any field, let X' be a projective and geometrically integral k-scheme and let Y' be a finite closed subscheme of X'. If f: Y'-> Y is a schematically dominant morphism between finite k-schemes and X is obtained by pinching X' along Y' via f, we describe the kernel (and, in certain cases, the cokernel) of the induced pullback map Br₁X -> Br₁X' between the corresponding algebraic (cohomological) Brauer groups of X and X'_solely_ in terms of the Brauer groups of the residue fields of Y and Y' and the Amitsur subgroups of X and X' in Br k. As an application, we compute the Brauer group of a projective and geometrically integral k-rational curve (with arbitrary singularities) over any field k. We also extend a well-known theorem of Roquette and Lichtenbaum to a class of singular curves over any local field.

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