116
N.M. Katz:
which extends V, whose
exponents
along each local irreducible branch
D i
of D lie in the strip 0__< R e (z)< 1. The pair (fgcan, V~an) is called the
quasicanonical
extension of ((~, V).
Although functorial in (~, V), the formation of the quasicanonical
extension does not commute with tensor product. However, given
(fg, V) and (~, V') there is a unique
horizontal
morphism
which prolongs the identity. Applying this with ~= f~ = ~, we find a
horizontal morphism
By functoriality, the algebra structure on ~
prolongs to a horizontal
morphism
Composing these last two maps, we obtain a horizontal multiplication
on ~an which extends the given algebra structure on
The locally free sheaf of algebras ~an on ~a~ corresponds to a finite
flat morphism of analytic spaces
f:
~(" ~ S an
which prolongs f: Y" ~ San. By GAGA, the morphism ~ comes from a
unique finite fiat morphism of proper Cschemes X~ S, and XIS> S
is a finite 6tale covering of S.
This construction is the desired inverse functor Etale (S an) ~ Etale (S).
Q.E.D.
References
0. Atiyah, M., Hodge, W.: Integrals of the second kind on an algebraic variety. Annals
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 Fall '11
 NormanKatz
 Algebra, Algebraic geometry, Functor, Cartier

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