Principal homogeneous space

We put Z = G/S faithfully flat since G is identified with a principal homogeneous space under S(a)z over Z. Thus we can use EGA IV 2.7.1 to see that the morphism G Z is of finite presentation. This allows us to use SGA Expose V Proposition 9.1 to conclude that Z is of finite type over Spec(Z). Finally, we leave it to the reader to show that Z is separated. 

Therefore we see that if F is left exact then c is bijective, hence is bijective, that is the action of tF on the fibres of F[Pg.179]

Principal homogeneous space, 8-5 Principal parts, sheaf of, 11 -8 Projective bundle, 1 -7 Projectively normal curve, 9-9 Prorepresentable functor, B-l Pullback of an extension, A-6 Pushout of an extension, A-6... 

Clearly G itself under multiplication has such a structure. Moreover, this is almost the only example, since for any x in X(k) the map g> gx is a bijection G - X preserving the G-action. The interest arises only from the seemingly minor fact that an X satisfying the definition may have X(k) empty. We do not however want the emptiness to extend too far, and we call X a principal homogeneous space (or torsor) for G only if X(S) 0 for some k- S faithfully flat. 

Let G be a fixed affine group scheme, A = k[G]. The structure of principal homogeneous space X for G is one to which descent theory applies. If kpf] = N, the structure on N is given by a multiplication N N - N and an action map N-> A N, the axioms are that certain diagrams commute, that N has a map to some faithfully flat k-algebra, and that a certain map... 

Theorem. Let k- S be faithfully flat, G an affine group scheme over k. The principal homogeneous spaces for G having a point in S are classified by H1(S/k, G). 

Theorem. Let G be an algebraic affine group scheme over a field k. Let X be a principal homogeneous space. Then k[X] is finitely generated, and X( ) is nonempty. 

Corollary. In this situation principal homogeneous spaces correspond to H k/k, G). 

Proof. Both of them classify principal homogeneous spaces. ... 

Let k be a ring, G an affine group scheme, X a principal homogeneous space. If k - fc[G] is faithfully flat, show k - k[X] is faithfully flat. 

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