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Unformatted text preview: 80 N.M. Katz: 6. Applications to the Hypergeometric Equation
6.0. Relations with Ordinary Differential Equations
(6.0.0) Let T a scheme, S a smooth Tscheme, and (M, V) a locally free
(gsmodule of finite rank with an integrable Tconnection. Recall that
the dual o f (M, V), noted (37/, P), is defined by
(6.0.0.1) ~ / = H omes (M, Cs) w ith the connection V defined by requiring that for local section m of M,
rh of ~/, and D of D er(S/T), we have
( 6.0.0.2) ( V(D)(m), th> + (m, V(D)(rh)>= O((m, th>). I teration of (6.0.0.2) gives, for every integer n > 1,
( 6.0.0.3) D'((m, rh))= ~ (~)((V(D))"i(m),(V(D))i(rh)).
i =0 T aking n =p in characteristic p, we find
(6.0.1) R emark. H ypotheses as in (6.0.0), if Tis a scheme of characteristic
p, then the pcurvature (cf. [24], 5.2) of (~/, V) is the n egative o f the transp ose of the pcurvature of (M, V):
(6.0.1.0) <Ov(D)(m), rh> + <m, r162
(D)(th)> =0. (6.0.2) Proposition. H ypotheses as in (6.0.0), suppose T is a reduced and
irreducible scheme of characteristic p. Let k denote its function field.
S uppose S is a smooth Tscheme which is irreducible, and denote by K its
f unction field. The following condition on (M, V) are equivalent. (6.0.2.0) ( M, V) has pcurvature zero.
(6.0.2.1) (K/I,~z) has pcurvature zero.
(6.0.2.2) The dimension over k . K p of (M  K) v is the rank of M.
(6.0.2.3) The dimension over k . K p of (A4  K) r is the rank of M.
P roof By (6.0.1), (6.0.2.0)r
so it suffices to show that
(6.0.2.0) r (6.0.2.2). For this, we form the cartesian diagram
S (p) o , S
(6.0.2.4) lg~,~
T e~ 1~
~T ...
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 Fall '11
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