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Formal fibers and birational extensions

Published online by Cambridge University Press:  22 January 2016

William Heinzer ,
Christel Rotthaus  and
Judith D. Sally
William Heinzer
Affiliation:
Department of Mathematics Purdue University, West Lafayette, IN 47907, USA
Christel Rotthaus
Affiliation:
Department of Mathematics Michigan State University, East Lansing, MI 48824, USA
Judith D. Sally
Affiliation:
Department of Mathematics Northwestern University, Evanston, IL 60201, USA
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Suppose (R, m) is a local Noetherian domain with quotient field K and m-adic completion Ȓ. It is well known that the fibers of the morphism Spec(Ȓ) ₒ Spec(R), i.e., the formal fibers of R, encode important information about the structure of R. Perhaps the most important condition in Grothendieck’s definition of R being excellent is that the formal fibers of R be geometrically regular. Indeed, a local Noetherian ring is excellent provided it is universally catenary and has geometrically regular formal fibers [G, (7.8.3), page 214]. But the structure of the formal fibers of R is often difficult to determine. We are interested here in bringing out the interrelatedness of properties of the generic formal fiber of R with the existence of certain local Noetherian domains C birationally dominating R and having C/mC is a finite R-module.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 131 , September 1993 , pp. 1 - 38
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

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