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Regular Overrings of Regular Local Rings

Judith Sally
Transactions of the American Mathematical Society
Vol. 171 (Sep., 1972), pp. 291-300
DOI: 10.2307/1996383
Stable URL: http://www.jstor.org/stable/1996383
Page Count: 10
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Regular Overrings of Regular Local Rings
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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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