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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A bracket power characterization
of analytic spread one ideals


Authors: L. J. Ratliff Jr. and D. E. Rush Jr.
Journal: Trans. Amer. Math. Soc. 352 (2000), 1647-1674
MSC (1991): Primary 13A15, 13B20, 13C10; Secondary 13H99
Published electronically: July 26, 1999
MathSciNet review: 1641107
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Abstract: The main theorem characterizes, in terms of bracket powers, analytic spread one ideals in local rings. Specifically, let $b_{1},\dots ,b_{g},x$ be regular nonunits in a local (Noetherian) ring $(R,M)$ and assume that $I$ $\subseteq $ $(xR)_{a}$, the integral closure of $xR$, where $I$ $=$ $(b_{1},\dots ,b_{g},x)R$. Then the main result shows that for all but finitely many units $u_{1},\dots ,u_{g}$ in $R$ that are non-congruent modulo $M$ and for all large integers $n$ and $k$ it holds that $I^{jn}$ $=$ $I^{[j]n}$ for $j$ $=$ $1,\dots ,k$ and $j$ not divisible by $char(R/M)$, where $I^{[j]}$ is the $j$-th bracket power $((b_{1}+u_{1}x)^{j}, \dots ,(b_{g}+u_{g}x)^{j},x^{j})R$ of $I$ $=$ $(b_{1}+u_{1}x, \dots ,b_{g}+u_{g}x,x)R$. And, conversely, if there exist positive integers $g$, $n$, and $k$ $\ge $ ${\binom{{n+g} }{{g}}}$ such that $I$ has a basis $\beta _{1},\dots ,\beta _{g} ,x$ such that $I^{kn}$ $=$ $({\beta _{1}}^{k},\dots ,{\beta _{g}}^{k},x^{k})^{n}R$, then $I$ has analytic spread one.


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Additional Information

L. J. Ratliff Jr.
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: ratliff@math.ucr.edu

D. E. Rush Jr.
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: rush@math.ucr.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02434-4
PII: S 0002-9947(99)02434-4
Keywords: Analytic spread, asymptotic prime divisor, binomial coefficient, bracket power of an ideal, essential prime divisor, integral closure of an ideal, local ring, Noetherian ring, persistent prime divisor, prenormal ideal, projectively equivalent ideals, Ratliff-Rush closure of an ideal, reduction of an ideal, superficial element
Received by editor(s): December 20, 1997
Published electronically: July 26, 1999
Article copyright: © Copyright 2000 American Mathematical Society