Dr. Katz DEq Homework Solutions 80

# Dr. Katz DEq Homework Solutions 80 - 80 N.M Katz 6...

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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 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 (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 p-curvature (cf. [24], 5.2) of (~/, V) is the n egative o f the transp ose of the p-curvature 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 T-scheme 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 p-curvature zero. (6.0.2.1) (K/I,~z) has p-curvature 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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