An elementary proof of the Briançon-Skoda theorem

Jacob Sznajdman[1]

  • [1] Mathematical Sciences, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg Sweden

Annales de la faculté des sciences de Toulouse Mathématiques (2010)

  • Volume: 19, Issue: 3-4, page 675-685
  • ISSN: 0240-2963

Abstract

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We give an elementary proof of the Briançon-Skoda theorem. The theorem gives a criterionfor when a function φ belongs to an ideal I of the ring of germs of analytic functions at 0 ∈ ℂ n ; more precisely, the ideal membership is obtained if a function associated with φ and I is locally square integrable. If I can be generated by m elements,it follows in particular that I min ( m , n ) ¯ ⊂ I , where J ¯ denotes the integral closure of an ideal J .

How to cite

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Sznajdman, Jacob. "An elementary proof of the Briançon-Skoda theorem." Annales de la faculté des sciences de Toulouse Mathématiques 19.3-4 (2010): 675-685. <http://eudml.org/doc/115874>.

@article{Sznajdman2010,
abstract = {We give an elementary proof of the Briançon-Skoda theorem. The theorem gives a criterionfor when a function $\phi $ belongs to an ideal $I$ of the ring of germs of analytic functions at $0\in \mathbb\{C\}^n$; more precisely, the ideal membership is obtained if a function associated with $\phi $ and $I$ is locally square integrable. If $I$ can be generated by $m$ elements,it follows in particular that $\overline\{I^\{\min (m,n)\}\}\subset I$, where $\overline\{J\}$ denotes the integral closure of an ideal $J$.},
affiliation = {Mathematical Sciences, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg Sweden},
author = {Sznajdman, Jacob},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {germs of holomorphic functions; integral closure; Briançon-Skoda theorem},
language = {eng},
number = {3-4},
pages = {675-685},
publisher = {Université Paul Sabatier, Toulouse},
title = {An elementary proof of the Briançon-Skoda theorem},
url = {http://eudml.org/doc/115874},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Sznajdman, Jacob
TI - An elementary proof of the Briançon-Skoda theorem
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2010
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - 3-4
SP - 675
EP - 685
AB - We give an elementary proof of the Briançon-Skoda theorem. The theorem gives a criterionfor when a function $\phi $ belongs to an ideal $I$ of the ring of germs of analytic functions at $0\in \mathbb{C}^n$; more precisely, the ideal membership is obtained if a function associated with $\phi $ and $I$ is locally square integrable. If $I$ can be generated by $m$ elements,it follows in particular that $\overline{I^{\min (m,n)}}\subset I$, where $\overline{J}$ denotes the integral closure of an ideal $J$.
LA - eng
KW - germs of holomorphic functions; integral closure; Briançon-Skoda theorem
UR - http://eudml.org/doc/115874
ER -

References

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  7. Lejeune-Jalabert (M.), Tessier (B.) and Risler (J.-J.).— Clôture intégrale des idéaux et équisingularité, Ann. Toulouse Sér. 6, 17 no. 4, p. 781-859, available at arXiv:0803.2369 (2008). Zbl1171.13005MR2499856
  8. Lipman (J.), Tessier (B.).— Pseudo-rational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28, p. 97-115 (1981). Zbl0464.13005MR600418
  9. Schoutens (H.).— A non-standard proof of the Briançon-Skoda theorem, Proc. Amer. Math. Soc. 131, p. 103-112 (2003). Zbl1005.13007MR1929029
  10. Skoda (H.).— Application des techniques L 2 à la théorie des idéaux d’une algébre de fonctions holomorphes avec poids, Ann. Sci. École Norm. Sup. (4) 5, p. 545-579 (1972). Zbl0254.32017MR333246

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