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

Mikhail Borovoi; Cyril Demarche; David Harari

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arXiv · 1203.5964

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

Mikhail Borovoi·Cyril Demarche·David Harari

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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.

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BibTeX RIS

Mikhail Borovoi, Cyril Demarche, David Harari. 2012-03-27. Complexes de groupes de type multiplicatif et groupe de Brauer non ramifi\'e des espaces homog\`enes. https://arxiv.org/abs/1203.5964

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