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 T-scheme, and (M, V) a locally free
(gs-module of finite rank with an integrable T-connection. Recall that
the dual o f (M, V), noted (37/, P), is defined by
(126.96.36.199) ~ / = 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
( 188.8.131.52) ( V(D)(m), th> + (m, V(D)(rh)>= O((m, th>). I teration of (184.108.40.206) gives, for every integer n > 1,
( 220.127.116.11) 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 p-curvature (cf. , 5.2) of (~/, V) is the n egative o f the transp ose of the p-curvature of (M, V):
(18.104.22.168) <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 T-scheme which is irreducible, and denote by K its
f unction field. The following condition on (M, V) are equivalent. (22.214.171.124) ( M, V) has p-curvature zero.
(126.96.36.199) (K/I,~z) has p-curvature zero.
(188.8.131.52) The dimension over k . K p of (M | K) v is the rank of M.
(184.108.40.206) The dimension over k . K p of (A4 | K) r is the rank of M.
P roof By (6.0.1), (220.127.116.11)r
so it suffices to show that
(18.104.22.168) r (22.214.171.124). For this, we form the cartesian diagram
S (p) o , S
T e~ 1~
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- Fall '11
- Vector Space, Complex number, hypergeometric equation, smooth T-scheme