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Dr. Katz DEq Homework Solutions 116

Dr. Katz DEq Homework Solutions 116 - 116 N.M Katz w hich...

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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 quasi-canonical extension of ((~, V). Although functorial in (~, V), the formation of the quasi-canonical 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 C-schemes 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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