76 N.M. Katz: zero. This implies that for every maximal ideal m of B such that B/m B has characteristic p~2;, the inverse image on B/m B of the mapping p(D) vanishes, i.e., all the matrix coefficients of p (D) lie in m. Because B is a finitely generated integral domain whose fraction field is of characteristic zero, it follows immediately from Noether's normalization theorem that for any infinite set 2" of prime numbers, the intersection of all maximal ideals of B with residue characteristic in 2" is reduced to zero. Thus the Kodaira-Spencer mapping (22.214.171.124) vanishes. This concludes the proof. 5.2. A Global Situation with a Group of Operators (5.2.0) Hypotheses as in (5.0.0), let G be a finite group, which acts as a group of S-automorphisms of X which are stable on D. Let g be the least common multiple of the orders of the elements of G, and let Ao= Z [l/g, ~g], where ~g is a primitive g-th root of unity. We suppose A ~ A o. (5.2.1) As is wellknown (cf. ), every irreducible representation of G
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class (cf., extendi ng scalars, Frobenius automorp hism, locally free Ao-module