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24
N.M. Katz:
of finite rank implies that for a suitable subring AI of
A, AocA 1 cA,
which is finitely generated over Z, the Almodule
E~'b(AO
is locally
free of finite rank, or what is the same, flat, since in any
case
it of finite
type over the noetherian ring A1.
The truth of this last assertion is a particularly simple case of (EGA IV,
11.2.6.1), in the notation of which Bo=Ao,
Mo=E~'b(Ao),
and the A~
are all the absolutely finitely generated subrings of A with A o c A~ ~ A.
(2.3.2.5)
Having reduced to the case in which S is the spectrum of a
noetherian ring, we may further assume that S is the spectrum of a
noetherian local ring (again because the question is local on S). We may
next suppose S to be the spectrum of a complete noetherian local ring
(by faithful flatness of the completion), and finally that S is the spectrum
of an artinian local ring. Let us first explain this last reduction step.
Suppose S = Spec(A), A a complete noetherian local ring with maximal
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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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