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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) (6.0.2.5) 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 (6.0.2.7) F* (F, (MY)) --~ M is an isomorphism. On the other hand, since the p-curvature may be interpreted as an (gs-homomorphism (6.0.2.8) ~: 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 (6.0.2.9) (m | K) v | K -* M
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Unformatted text preview: kKP is an isomorphism. Thus it remains to prove only that (6.0.2.9) is an isomorphism if and only if its source and target have the same dimension. This is indeed the case, because (6.0.2.9) 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 (6.0.2.7)) (6.0.3.0) 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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