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 $\mathrm{UPic}(\overline{X})$. It is a complex of length 2 which combines the Picard group $\mathrm{Pic}(\overline{X})$ and the group $U(\overline{X}):=\overline{k}{\overline{X}]}^{\times }/{\overline{k}}^{\times }$, where $\overline{k}$ is a fixed algebraic closure of k and $\overline{X}=X{\times }_k\overline{k}$. For a connected linear k-group G we compute the complex $\mathrm{UPic}(\overline{G})$ (up to a quasi-isomorphism) in terms of the algebraic fundamental group ${\pi }_1(\overline{G})$. 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).