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Unformatted text preview: vertible function g with 09 = dg/g. Proof (7.1.3.1)r by (7.1.2.0). To see that (7.1.3.1)r (7.1.3.3), recall that by Cartier's theorem (cf. (6.0.3) and [24], Theorem 5.1), ((gs, 17j has pcurvature zero if and only if (9 s is spanned as (9 smodule by the subsheaf of germs of horizontal functions. Thus (7.1.3.1) is true if and only if there exists locally on S an invertible section of (gs, f, with 0= Vo,(f)=df+f09, or equivalently (taking g=f1), if and only if to is locally logarithmic. Q.E.D. (7.1.4) Remark. In Cartier's original "operator" (cf. [2]), the absolute Frobenius Fab s: T, T was an isomorphism; T was, in fact, the spectrum of a perfect field. Then a: S~P~ S was also an isomorphism, and the original Cartier operation rgorigi,a~ was defined as an additive isomor phism (7.1.4.0) (~original: ~ ~ ' ~'~/T which satisfied (7.1.4.1) C~original(fP09)=f; ~original (09)...
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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