Extended Picard complexes for algebraic groups and homogeneous spaces

For a smooth geometrically integral algebraic variety X over a field k of characteristic 0, we define the extended Picard complex UPic ( over(X, ̄) ). It is a complex of length 2 which combines the Picard group Pic ( over(X, ̄) ) and the group U ( over(X, ̄) ) : = over(k, ̄) [ over(X, ̄) ]^× / over(k, ̄)^×, where over(k, ̄) is a fixed algebraic closure of k and over(X, ̄) = X ×_k over(k, ̄). For a connected linear k-group G we compute the complex UPic ( over(G, ̄) ) (up to a quasi-isomorphism) in terms of the algebraic fundamental group π₁ ( over(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).