Extended Picard complexes for algebraic groups and homogeneous spaces

Abstract

For a smooth geometrically integral algebraic variety X over a field k of characteristic 0, we define the extended Picard complex UPic ( X ¯ ) . It is a complex of length 2 which combines the Picard group Pic ( X ¯ ) and the group U ( X ¯ ) : = k ¯ [ X ¯ ] × / k ¯ × , where k ¯ is a fixed algebraic closure of k and X ¯ = X × k k ¯ . For a connected linear k-group G we compute the complex UPic ( G ¯ ) (up to a quasi-isomorphism) in terms of the algebraic fundamental group π 1 ( G ¯ ) . We obtain similar results for a homogeneous space X of a connected k-group G. To cite this article: M. Borovoi, J. van Hamel, C. R. Acad. Sci. Paris, Ser. I 342 (2006).