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Dr. Katz DEq Homework Solutions 83

# Dr. Katz DEq Homework Solutions 83 - Let(M V be as in(6.0.4...

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Algebraic Solutions of Differential Equations 83 Let eo .... , ~,_ 1 denote the dual basis of the dual module AS/. A section (6.0.4.1) ~ fi ei, f/ local sections of (9 s i=o is horizontal for the dual connection V on M/f and only if its coefficients satisfy dfi dX =f/+x for O<i<n-2 (6.0.4.2) dfn- 1 ,-1 Y a, dX i=o Thus (6.0.4.3) The projection A7/-*(9s defined by ~Jiki--*fo induces an isomorphism between (A~/) ~, the sheaf of germs of horizontal sections of M, and the sheaf of germs of sections of (9 s which are annihilated by V2- -F a, -Z2- i=0 If T has characteristic p, this isomorphism is F- 1 ((gs~p,)_linear. Combining (6.0.4.3) and (6.0.2), we find (6.0.5) Proposition. Let T be a reduced and irreducible scheme of charac- teristic p, S an irreducible scheme which is dtale over A~T by means of a section X of(9 s. Let k (resp. K) denote the function field of T (resp. S).
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Unformatted text preview: Let (M, V) be as in (6.0.4). Then (M, V) has p-curvature zero if and only if the field K contains n = rank M solutions of the ordinary differential equation (6.0.5.1) (f)= ~ ai (f) i=0 which are linearly independent over the subfield k. K p. (6.0.6) Suppose further that k is a perfect field (then kKP=KP). For any discrete valuation v of K, let v o denote the induced valuation of K p. Clearly the ramification index e(V/Vo) (= the index of the value groups) is p. As K is a p-dimensional vector space over K p, it follows that if t is a uniformizing parameter in K vor v, the elements 1, t . ... , t p-I form a basis of K over K p, and thus provide an isomorphism of KP-vector spaces (6.0.6.0) K--- KP(~ . .. G K p. ptimes In terms of this isomorphism, the v-adic topology on K is just the p-fold product of the vo-adic topology on KP; 6*...
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