Journal für die Reine und Angewandte Mathematik (Crelle's Journal) vol:627 pages:53-82
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 (k) over bar [X](x)/(k) over bar (x) and whose first cohomology is Pic((X) over bar), where (k) over bar is a fixed algebraic closure of k and X is obtained from X by extension of scalars to (k) over bar. 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 pi(1)(G). As an application we compute the elementary obstruction for such X.