Unformatted text preview: on q/i. Thus we have: (7.2.1) Proposition. The group of isomorphism classes of invertible sheaves on S with Tconnections (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 Tconnection 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 (gsmodule with integrable Tconnection, given by the data (Col,fig) on an open covering ~ of S, its pcurvature (7.2.2.0) ~O: Der(S/T) ~ End~s(L~')_~ (gs is the plinear 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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 Fall '11
 NormanKatz
 De Rham cohomology, Invertible Sheaves, compatib ility

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