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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. pCurvature and the Cartier Operation;
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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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