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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
DOI: https://doi.org/10.1090/S0002-9939-02-06556-5
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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  • 1. M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 177-291. MR 38:344
  • 2. -, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23-58. MR 42:3087
  • 3. M. Artin and C. Rotthaus, A structure theorem for power series rings, Algebraic Geometry and Commutative Algebra: in honor of M. Nagata (Tokyo), vol. I, Kinokuniya, 1988, pp. 35-44. MR 90b:14006
  • 4. J. Becker, J. Denef, L. van den Dries, and L. Lipshitz, Ultraproducts and approximation in local rings I, Invent. Math. 51 (1979), 189-203. MR 80k:14009
  • 5. J. Briançon and H. Skoda, Sur la clôture intégrale d'un idéal de germes de fonctions holomorphes en un point de $C^n$, C. R. Acad. Sci. Paris 278 (1974), 949-951. MR 49:5394
  • 6. J. Denef and H. Schoutens, On the decidability of the existential theory of $\mathbb{F}_p[[t]]$, Valuation Theory and its Applications (F.-V. Kuhlmann, S. Kuhlmann, and M. Marshall, eds.), Fields Institute Communications Series, Amer. Math. Soc., to appear, preprint on URL address http://www.math.ohio-state.edu/~schoutens/ExistentialTheory.ps.
  • 7. M. Fried and M. Jarden, Field arithmetic, Springer-Verlag, 1986. MR 89b:12010
  • 8. M. Hochster and C. Huneke, Tight closure in equal characteristic zero, preprint available on URL address http://www.math.lsa.umich.edu/~hochster/tcz.ps.Z, 2000.
  • 9. W. Hodges, Model theory, Cambridge University Press, Cambridge, 1993. MR 94e:03002
  • 10. C. Huneke, Tight closure and its applications, Conference Board in Math. Sciences, vol. 88, Amer. Math. Soc., 1996. MR 96m:13001
  • 11. J. Lipman and A. Sathaye, Jacobian ideals and a theorem of Briançon-Skoda, Michigan Math. J. 28 (1981), 199-222. MR 83m:13001
  • 12. J. Lipman and B. Teissier, Pseudo-rational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), 97-116. MR 82f:14004
  • 13. H. Matsumura, Commutative algebra, W.A. Benjamin, 1970. MR 42:1813; second ed. MR 82i:13003
  • 14. H. Schoutens, Finitistic power series, preprint available on URL address http://www.math.ohio-state.edu/~schoutens/, 2001.
  • 15. -, Lefschetz principle applied to symbolic powers, preprint available on URL address http://www.math.ohio-state.edu/~schoutens/SymbolicPowersLefschetz.ps, 2001.
  • 16. -, Non-standard tight closure, preprint on URL address http://www.math.ohio-state.edu/ ~schoutens/NonStandardTightClosure.ps, 2001.
  • 17. -, Reduction modulo $p$ of power series with integer coefficients, preprint available on URL address http://www.math.ohio-state.edu/~schoutens/IntegerPowerSeries.ps, 2001.
  • 18. C. Wall, Lectures on $C^\infty$ stability and classification, Proceedings of Liverpool Singularities-Symposium I, Lect. Notes in Math., vol. 192, Springer-Verlag, 1971, pp. 178-206. MR 44:2244
  • 19. O. Zariski and P. Samuel, Commutative algebra, Van Nostrand, Princeton, 1960. MR 22:11006

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Additional Information

Hans Schoutens
Affiliation: Department of Mathematics, 100 Math Tower, Ohio State University, Columbus, Ohio 43210
Email: schoutens@math.ohio-state.edu

DOI: https://doi.org/10.1090/S0002-9939-02-06556-5
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

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