Journal of the Ramanujan Mathematical Society

Volume 39, Issue 2, June 2024 pp. 193–210.

The algebraic Brauer group of a pinched variety

Authors: Cristian D. González-Avilés
Author institution:Departamento de Matemáticas, Universidad de La Serena, Cisternas 1200, La Serena 1700000, Chile

Summary: 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 ψ : {Y} → Y is a schematically dominant morphism between finite k-schemes and X is obtained by pinching {X} along {Y} via ψ, we describe the kernel (and, in certain cases, the cokernel) of the induced pullback map Br{1} X → Br{1} {X} 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 algebraic Brauer group of a projective and geometrically integral k-scheme X with a finite non-normal locus whose normalization X{N} is k-isomorphic to ℙ{k}{dim X}. If k is a local field and X is a projective and geometrically integral k-curve such that X{N} is {smooth}, then we show that the order of the Amitsur subgroup of X in Br k is the index of X, i.e., the least positive degree of a 0-cycle on X. This statement generalizes a well-known theorem of Roquette and Lichtenbaum, who obtained the above conclusion when k is a p-adic field and X is smooth over k.

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