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Dr. Katz DEq Homework Solutions 106

# Dr. Katz DEq Homework Solutions 106 - vertible function g...

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106 N.M. Katz: As 09 is closed, it may be written (cf. (2.2.1)) (7.1.2.7) 09=~ F*(ai) se~ -1 dsi+dg; al ..... a,,geF(S, (gs). As both sides of the asserted equality (7.1.2.6), regarded as function of the closed one-form 09, are F-l(t~s~p~)-linear and annihilate exact one- forms, we are reduced to the case (7.1.2.8) 09 -- s~ -1 ds i . Then (cf. (2.2.1)) (7.1.2.9) cg (09) = d (o'* (si)), and (7.1.2.6) becomes (7.1.2.10) F* ((dtr* (si), tr* (D))) = (s p- 1 d s i , D p) - D p- 1((s~-1 dsl, O)), or equivalently (7.1.2.11) (D(si))P=s~-I DP(si)-DP-I(s~-I D(si)) which is none other than Hochschild's identity (3.5.2.6). Q.E.D. (7.1.3) Corollary. Hypotheses as in 7.1, the following conditions on a closed one-form 09e F(S, f21s/7.) are equivalent. (7.1.3.1) The connection Vo, on (9 s has p-curvature zero (7.1.3.2) cg(09)= tr* (09) (7.1.3.3) 09 is locally logarithmic, i.e., locally on S there exists an in-
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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 p-curvature zero if and only if (9 s is spanned as (9 s-module 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=f-1), 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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