Complexes de groupes de type multiplicatif et groupe de Brauer non ramifié des espaces homogènes

Details Complexes de groupes de type multiplicatif et groupe de Brauer non ramifié des espaces homogènes Complexes de groupes de type multiplicatif et groupe de Brauer non ramifié des espaces homogènes

2012-03-27

arxiv.org/abs/1203.5964v1

Mathematics

Algebraic Geometry

42 pages

Scientific paper

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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