Extended equivariant Picard complexes and homogeneous spaces

Let k be a field of characteristic 0 and let k̄ be a fixed algebraic closure of k. Let X be a smooth geometrically integral k-variety; we set X̄ = X × _kk̄ and denote by K(X̄) the field of rational functions on X̄. In [BvH2] we defined the extended Picard complex of X as the complex of Gal(k̄/k)-modules, where K(X̄) ^×/k̄ ^× is in degree 0 and Div(X̄) is in degree 1. We computed the isomorphism class of UPic(Ḡ) in the derived category of Galois modules for a connected linear k-group G. Here we compute the isomorphism class of UPic(X̄) in the derived category of Galois modules when X is a homogeneous space of a connected linear k-group G with Pic(Ḡ)=0. Let x̄ ∈ X(K̄) and let H̄ denote the stabilizer of x̄ in Ḡ. It is well known that the character group X(H̄) of H̄ has a natural structure of a Galois module. We prove that, in the derived category, where res is the restriction homomorphism. The proof is based on the notion of the extended equivariant Picard complex of a G-variety.