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Dr. Katz DEq Homework Solutions 104

# Dr. Katz DEq Homework Solutions 104 - 104 N.M Katz which is...

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104 N.M. Katz: which is invertible modulo n, we have either 1 >(fa)>(fc)>(fb)>0 (6.9.4.1) or l>(fb)>(fc)>(fa)>0. Proof This follows from 6.2 and 6.6.0. (6.9.5) Remark. (Interpretation of (6.9.4)). Given three distinct roots of unity 41, 42, ~3 in C all distinct from 1, we say that 41 and 4z separate ~a and 1 if, in marching counterclockwise around the unit circle, starting at i, we encounter either ~1 or ~2 but not both before we encounter 43. Let 4./.=exp (2rti a ) (6.9.5.0) 4b/,=exp (2~ri b) ~c/. = exp (2re i ~-) - The condition (6.9.4.1) is "simply" that these three roots of unity are distinct from each other and from 1, and that for every automorphism a of C, (6.9.5.1) (4,/,)" and (4b/,)" separate (4c/,) ~ 1. 7. p-Curvature and the Cartier Operation;
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