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Dr. Katz DEq Homework Solutions 107

Dr. Katz DEq Homework Solutions 107 - on q/i Thus we...

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Algebraic Solutions of Differential Equations 107 In our notation, the relation between <gorigi,al and cr is just (7.1.4.2) <r = (tr- 1), o <g. 7.2. Let T be an arbitrary scheme, S a smooth T-scheme. Let ~,e be an invertible sheaf on S, given by transition function f~j with respect to an open covering q/i of S. Then giving a T-connection 17 on ~ is equivalent to giving, for each q/i, a one form tDiG_r(dlli, ols/T) subject to the compati- bility (7.2.0.0) co i - a~j = dfij/fij. The connection is integrable if and only if each o9 i is closed. If (s 17) and (~', 17') are two invertible sheaves with connection, both given with respect to the same open covering q/i of S by data (col,f0 and (<n'i,f/~) as above, an isomorphism between (s 17) and (~', 17') is just the giving of invertible functions gi~F(qli, C~') subject to the conditions (7.2.0.1) f~i =f~(gi/gj) on q/i n q/~ CO i- CO' i = dgi/g i
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Unformatted text preview: on q/i. Thus we have: (7.2.1) Proposition. The group of isomorphism classes of invertible sheaves on S with T-connections (Z.g~', 17) (under tensor product) is H 1 (S, (9* d log ~, ~'~Is/T) " The group of isomorphism classes of invertible sheaves on S with integrable T-connection is H I (S, f2*/T) where 12~*/T denotes the multiplicative de Rham complex (7.2.1.0) (9* a'o~ , f2~/T d---~ f2Z/T a , . ... As a corollary of (7.1.2), we have (7.2.2) Proposition. If T has characteristic p, and (~, 17) is an invertible (gs-module with integrable T-connection, given by the data (Col,fig) on an open covering ~ of S, its p-curvature (7.2.2.0) ~O: Der(S/T) ~ End~s(L~')_~ (gs is the p-linear mapping given locally by (7.2.2.1) ~O(D)=F*((tr*(ogi)-cs a*(D))) over q/i- [This formula has a global meaning, because, by (7.1.3.3), ~r* (~o0- ~(o~)-,r* (~o~) + ~(co~) (7.2.2.2) = a* (dfij/fij) - ~g(dfiJfij) = 0]....
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