# Is there a reference that treats principal homogeneous spaces for (say) finite type $k$-groups using schemes?
I was wondering if anyone could recommend a reference that discusses principal homogeneous spaces for general finite type group schemes over a field $k$ entirely in the language of schemes (or even just principal homogeneous spaces for arbitrary group schemes, though I'm not sure how much people care about such things). What I'm particularly interested in is a proof (using the modern language) of the surjectivity of the map sending the isomorphism classes of principal homogeneous spaces for $G$ to $H^1(k,G(k^{sep}))$, where $G$ is a group variety over $k$ (by which I mean a separated, finite type, geometrically integral $k$-group). I'm pretty certain most aspects of the argument for this bijection (say as given in the paper Principal Homogeneous Spaces Over Abelian Varieties of Lang and Tate) can be translated fairly naturally into scheme language, but for surjectivity they invoke a result of Weil that I would like to avoid. I know also that this is treated in Silverman, but not using schemes.