A non-standard proof of the Briançon-Skoda theorem

Schoutens, Hans

doi:10.1090/S0002-9939-02-06556-5

A non-standard proof of the Briançon-Skoda theorem
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Proc. Amer. Math. Soc. 131 (2003), 103-112 Request permission

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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