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Here, a Galois A-object is exactly what geometers mean by a principal homogeneous space (PHS) for Spec A over Spec R, the Extzar refers to extensions of sheaves in the Zariski topology over R, and AD is the linear R-dual of A. We wish to give a new, short proof of this theorem having two advantages: It applies even when Spec R is replaced by an arbitrary prescheme, and it explains why only the Zariski topology is needed. The price paid is that a certain amount of machinery is used, the proof we give being less explicit than the original one. If 7r: Y-)X is an affine morphism of preschemes, we shall say that Y is locally a projective module over X if and only if for every affine open U in X, r(7r-(U), oy) is a projective r(u, ox) module. As an example, if X is noetherian and Y is finite and flat over X, then Y is locally a projective module over X.

Principal homogeneous spaces for finite group schemes