Algebraic Solutions of Differential Equations 17 morphism Fab~: S-~S (so on the ring level, Fabs is just "raising to the p-th power"). Thus by construction YtP~ sits in a cartesian diagram (184.108.40.206) The pair of morphisms I ~(P) = Fa*bs(~) [~ S r~b~ ~S Fab s : y----~ y 7z : Y---~ S defines a morphism F: Y--~ Y~P~, the relative Frobenius, which fits into a commutative diagram y f ~ y~p~ ~ ~ y r ~ y(p) ~(P) in which F. ~ is F,b s: Y(P) ~ Y(P) and a- F is F~b s: Y---~ Y. Intuitively, F raises the "vertical coordinates" to the p-th power, and e raises the "S coordinates" to the p-th power. (2.0.2) Consider now the special case Y=X is a smooth S-scheme, say of relative dimension n. Then: (220.127.116.11) F,(Cx) is a locally free (gxc,~ module of rank pn; indeed if
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