Regular overrings of regular local rings

Sally, Judith

doi:10.1090/S0002-9947-1972-0309929-3

Regular overrings of regular local rings
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by Judith Sally PDF

Trans. Amer. Math. Soc. 171 (1972), 291-300 Request permission

Erratum: Trans. Amer. Math. Soc. 213 (1975), 429.

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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