# 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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