Extended Picard complexes and linear algebraic groups

For a smooth geometrically integral variety $X$ over a field $k$ of characteristic 0, we introduce and investigate the extended Picard complex $UPic(X)$. It is a certain complex of Galois modules of length 2, whose zeroth cohomology is $\bar{k}[X]^*/ \bar{k}^*$ and whose first cohomology is $Pic(\bar{X})$, where $\bar{k}$ is a fixed algebraic closure of $k$ and $\bar{X}$ is obtained from $X$ by extension of scalars to $\bar{k}$. When $X$ is a $k$-torsor of a connected linear $k$-group $G$, we compute $UPic(X)=UPic(G)$ (in the derived category) in terms of the algebraic fundamental group $π_1(G)$. As an application we compute the elementary obstruction for such $X$.

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