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 (22.214.171.124) 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: (126.96.36.199) F,(Cx) is a locally free (gxc,~ module of rank pn; indeed if
This is the end of the preview.
access the rest of the document.
Complex number, De Rham cohomology, fabs, Epimorphism, p-th power