Complexes de groupes de type multiplicatif et groupe de Brauer non ramifi\'e des espaces homog\`enes

TL;DR

This paper provides formulas for the algebraic Brauer group of homogeneous spaces under certain algebraic groups, extending to positive characteristic and cases without smooth compactifications, with implications for arithmetic geometry.

Contribution

It introduces new formulas for the algebraic Brauer group of homogeneous spaces with specific stabilizer structures, applicable in various characteristic settings.

Findings

Formulas for the algebraic Brauer group in characteristic zero.

Extensions of formulas to positive characteristic, including finite and global fields.

Results applicable even without smooth compactifications, under connected stabilizers.

Abstract

Let k be a field, G a smooth connected linear algebraic group and X a homogeneous space of G over k, such that the geometric stabilizers are extensions of a smooth group of multiplicative type by a smooth connected characterfree group. If k has characteristic zero and if X^c is a smooth compactification of X over k, we obtain a formula for the algebraic Brauer group of X^c. Several variants are obtained in positive characteristic p, including the finite field case and the global field case, where the formulae describe the prime-to-p part of the algebraic unramified Brauer group of X, without assuming the existence of a smooth compactification of X. Moreover, assuming that stabilizers are connected, then our formulae hold for the prime-to-p part of the whole unramified Brauer group.