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 (22.214.171.124) (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 (126.96.36.199) ~O: Der(S/T) ~ End~s(L~')_~ (gs is the p-linear mapping given locally by (188.8.131.52) ~O(D)=F*((tr*(ogi)-cs a*(D))) over q/i- [This formula has a global meaning, because, by (184.108.40.206), ~r* (~o0- ~(o~)-,r* (~o~) + ~(co~) (220.127.116.11) = a* (dfij/fij) - ~g(dfiJfij) = 0]....
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- Fall '11
- De Rham cohomology, Invertible Sheaves, compatib ility