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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Regular overrings of regular local rings

Author: Judith Sally
Journal: Trans. Amer. Math. Soc. 171 (1972), 291-300
MSC: Primary 13H05
Erratum: Trans. Amer. Math. Soc. 213 (1975), 429.
MathSciNet review: 0309929
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Abstract: The local factorization theorem of Zariski and Abhyankar characterizes all $ 2$-dimensional regular local rings which lie between a given $ 2$-dimensional regular local ring $ R$ and its quotient field as finite quadratic transforms of $ R$. This paper shows that every regular local ring $ R$ of dimension $ n > 2$ has infinitely many minimal regular local overrings which cannot be obtained by a monoidal transform of $ R$. These overrings are localizations of rings generated over $ R$ by certain quotients of elements of an $ R$-sequence. Necessary and sufficient conditions are given for this type of extension of $ R$ to be regular.

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Keywords: Regular local ring, overring, valuation domain, monoidal and quadratic transforms of a regular local ring, dimension formula, $ R$-sequence, locality, analytically irreducible domain
Article copyright: © Copyright 1972 American Mathematical Society