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Dr. Katz DEq Homework Solutions 81

Dr. Katz DEq Homework Solutions 81 - kKP is an isomorphism...

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Algebraic Solutions of Differential Equations 81 and "decorate" it with the relative Frobenius F: S--~ Stp) ( S F ~, S(p) a ~ S F ~, S(p) T f~bs ~T The corresponding diagram of function fields is K, ,k. KPt .... K+__ \k.K p o 61 k~ xp~x k By Cartier's theorem (cf. [24], 5.1), (M, V) has p-curvature zero if and only if the canonical morphism ( F* (F, (MY)) --~ M is an isomorphism. On the other hand, since the p-curvature may be interpreted as an (gs-homomorphism ( ~: M ~ F*b~ ((2~S/T) |162 M between locally free modules, it vanishes if and only if it vanishes over the generic point of S. Using Cartier's theorem over the generic point, the p-curvature vanishes there if and only if the canonical map of K-vector spaces ( (m | K) v | K -* M
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Unformatted text preview: kKP is an isomorphism. Thus it remains to prove only that ( is an isomorphism if and only if its source and target have the same dimension. This is indeed the case, because ( is always injective; in fact we have the apparently more general (6.0.3) Proposition. Let T be a scheme of characteristic p, S a smooth T-scheme, and (M, V) a quasi-coherent (gs-module with integrable T- connection. The canonical mapping ( cf the diagram ( ( F, (M v) | (gs = F* (F, (MY)) --~ M is always injective. Proof The question is local on S, so we may assume S is 6tale over A~ via X1, . .., X,. Then (9 s is a free F-X(~s~,~)-module with basis the monomials X wd~f'n xW'. .. X w" having 0< Wi <p- 1. Thus we must show 6 Inventiones math., Vol. 18...
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