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Seminormal rings and weakly normal varieties

Published online by Cambridge University Press:  22 January 2016

John V. Leahy
Affiliation:
Institute of Theoretical Sciences and Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Marie A. Vitulli
Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403
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Extract

In the late sixties and early seventies the operation of weak normalization was formally introduced first in the case of analytic spaces and later in the abstract scheme setting (cf. [6] & [4]). The notion arose from a classification problem. An unfortunate phenomenon in this area occurs when one tries to parametrize algebraic objects associated with a space by an algebraic variety; the resulting variety is, in general, not uniquely determined and may, for example, depend on the choice of coordinates. Under certain conditions one does know that the normalization of the parameter variety is unique. The price one pays for passing to the normalization is that often this variety no longer parametrizes what it was intended to; one point on the original parameter variety may split into several in the normalization. This problem is avoided if one passes instead to the weak normalization of the original variety. One then obtains a variety homeomorphic to the original variety with universal mapping properties that guarantee uniqueness.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

References

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