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Algebraic Solutions of Differential Equations
17
morphism Fab~:
S~S
(so on the ring level, Fabs is just "raising to the
pth power"). Thus by construction YtP~ sits in a cartesian diagram
(2.0.1.1)
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 pth power, and e raises the
"S coordinates" to the pth power.
(2.0.2)
Consider now the special case Y=X is a smooth Sscheme, say
of relative dimension n. Then:
(2.0.2.1)
F,(Cx)
is a locally free (gxc,~ module of rank pn; indeed if
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 Fall '11
 NormanKatz

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