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A non-standard proof of the Briançon-Skoda theorem

Author: Hans Schoutens
Journal: Proc. Amer. Math. Soc. 131 (2003), 103-112
MSC (2000): Primary 13A35, 13B22, 12L10
Published electronically: May 29, 2002
MathSciNet review: 1929029
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Abstract: Using a tight closure argument in characteristic $p$ and then lifting the argument to characteristic zero with the aid of ultraproducts, I present an elementary proof of the Briançon-Skoda Theorem: for an $m$-generated ideal $\mathfrak{a}$ of ${\mathbb C}[[{X_1,\dots,X_n}]]$, the $m$-th power of its integral closure is contained in $\mathfrak{a}$. It is well-known that as a corollary, one gets a solution to the following classical problem. Let $f$ be a convergent power series in $n$ variables over $\mathbb C$ which vanishes at the origin. Then $f^n$ lies in the ideal generated by the partial derivatives of $f$.

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Additional Information

Hans Schoutens
Affiliation: Department of Mathematics, 100 Math Tower, Ohio State University, Columbus, Ohio 43210

Keywords: Brian\c{c}on-Skoda, tight closure, ultraproducts
Received by editor(s): April 6, 2001
Received by editor(s) in revised form: September 3, 2001
Published electronically: May 29, 2002
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2002 American Mathematical Society