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- W2022153957 abstract "1. Let L be a noetherian local ring with maximal ideal M. To each ideal H of L there corresponds an ideal H., the integral closure or v-completion of H, which consists of those elements x in L that depend integrally upon H in the sense that an equation, xn+a,xn-1 + * * * +an=0, in which aiGHi, i=1, 2, . .. , n, holds. Any ideal H such that Ha =H is called a v-complete ideal. It is well known [3] that (Ha)a =Ha for all ideals H. In this note we study the sequence {Qn } (Qn = (Mn)a) of the v-completions of the powers of the maximal ideal M in the case where L is a two dimensional local domain that is integrally closed in its quotient field F. Subject to certain finiteness restrictions explained below it is shown that there is an ideal X generated by a set of parameters and an integer s such that for each positive integer t, Q8+t = Q8Xt. By means of this result it is shown that the length X(Qn) of the M-primary ideal Qn is given by a polynomial in n when n is large. Finally, it is shown that if L is the local ring at a point P of a normal algebraic surface VO, if 1l is the transform of VO defined by a quadratic transformation with center P, and if 'U6 is a derived normal model of cU associated with the character of homogeneity 3, then X(O, 2nb) -X(cU, n) =X(Qn), where X is the Hilbert characteristic function. In particular, the constant term of the polynomial X(Qn) measures the change in the relative arithmetic genus of U that is induced by the quadratic transformation. Some of the ideas and techniques used below are related to those employed by Rees in his study of pseudo-valuations [6, and related papers]. They go back ultimately, however, to Zariski's work on the reduction of singularities. The author expresses his thanks to the referee for several helpful suggestions." @default.
- W2022153957 created "2016-06-24" @default.
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- W2022153957 date "1960-03-01" @default.
- W2022153957 modified "2023-09-26" @default.
- W2022153957 title "Complete ideals in local rings" @default.
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- W2022153957 doi "https://doi.org/10.1090/s0002-9939-1960-0111753-9" @default.
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