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We give two proofs, one here and the second a little later. The first proof generalizes the proof we gave in the classical case. We consider not only the adjoint representation of X in the tangent space at the identity e, or in the space of differentials at e, but in each of the spaces (OS,/2g,) where Og,, is the local ring of X at e and 97tg,, its maximal ideal. For x e X, let CC: X -+ X be defined by CZ(y) = x y x-1, so that CZ(e) = e. Then C. 1 of the local ring. This induces a set theoretic map y: X a Aut(OS,,/TZ4 X's , x Q* and if we put on the latter group the natural structure of an algebraic variety (viz.

The conditions (i)-(iii) are evidently fulfilled with m instead of m + 1. Suppose then that m = - 1, that is, that {KP, aP} have been defined for p > 0 satisfying (i)-(iii). We then replace K° by K°/ker a° n Ker 00, and we take 00: K° - CO and a°: K° -> K1 to be the induced mappings. Putting KP = 0 for p < 0, we get a complex 0>K°>Kl>K2->K3->... K"-+ 0 and a homomorphism 0: K' > C' which by construction induces isomorphisins in cohomology. We have only to check that K° is A-flat when all the OP are A-flat.

HP(X,,, FFy) is a constant function, (i) Rpf. (Jr) is a locally free sheaf & on Y, and for all y e Y, the (ii) natural map - ®m y k(y) > HP(X, ,,) is an isomorphism. If these conditions are fulfilled, we have further that HP-1(Xy, Fr,) RP-' f* (F) ®aYk(y) is an isomorphism for all y e Y. PROOF. Again assume Y affine, S' as in the proposition. (ii) (i) is obvious. To prove (i) LEMMA 1. If Y is reduced and.. (ii), we need two lemmas. a coherent sheaf on Y such that dims[ ° ®0Yk(y)] = r, all y e Y, then JF is locally free of rank ronY.

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