On the Briançon-Skoda theorem on a singular variety

Andersson, Mats; Samuelsson, Håkan; Sznajdman, Jacob

Annales de l'Institut Fourier, Tome 60 (2010) no. 2, pp. 417-432.

Soit $Z$ un germe d’un espace analytique réduit de dimension pure. Nous donnons une démonstration analytique du théorème de Briançon-Skoda pour l’anneau local ${𝒪}_Z$. Ce résultat a déjà été démontré par Huneke en utilisant des méthodes algébriques. Nous obtenons également un résultat beaucoup plus fort pour les idéaux engendrés par peu d’éléments.

Let $Z$ be a germ of a reduced analytic space of pure dimension. We provide an analytic proof of the uniform Briançon-Skoda theorem for the local ring ${𝒪}_Z$; a result which was previously proved by Huneke by algebraic methods. For ideals with few generators we also get much sharper results.

Classification : 32C30, 32A27, 13A05
Keywords: Briançon-Skoda theorem, analytic space, residue current
Mot clés : théorème de Briançon-Skoda, espace analytique, courant résiduel

[1] Andersson, Mats Coleff-Herrera currents, duality, and Noetherian operators Preprint Gothenburg (2009), available at arXiv:0902.3064

[2] Andersson, Mats A residue criterion for strong holomorphicity Ark. Mat., (to appear) available at arXiv:0711.2863

[3] Andersson, Mats Integral representation with weights. I, Math. Ann., Volume 326 (2003) no. 1, pp. 1-18 | DOI | MR | Zbl

[4] Andersson, Mats Residue currents and ideals of holomorphic functions, Bull. Sci. Math., Volume 128 (2004) no. 6, pp. 481-512 | DOI | MR | Zbl

[5] Andersson, Mats Explicit versions of the Briançon-Skoda theorem with variations, Michigan Math. J., Volume 54 (2006) no. 2, pp. 361-373 | DOI | MR | Zbl

[6] Andersson, Mats Integral representation with weights. II. Division and interpolation, Math. Z., Volume 254 (2006) no. 2, pp. 315-332 | DOI | MR | Zbl

[7] Andersson, Mats; Götmark, E. Explicit representation of membership of polynomial ideals Preprint Mittag-Leffler (2008), available at arXiv:0806.2592

[8] Andersson, Mats; Samuelsson, Håkan Koppelman formulas and the $\overline{\partial }$-equation on an analytic space Preprint Mittag-Leffler (2008), available at arXiv:0801.0710

[9] Andersson, Mats; Wulcan, Elizabeth Decomposition of residue currents J. reine angew. Math., (to appear), available at arXiv:0710.2016

[10] Andersson, Mats; Wulcan, Elizabeth Residue currents with prescribed annihilator ideals, Ann. Sci. École Norm. Sup. (4), Volume 40 (2007) no. 6, pp. 985-1007 | Numdam | MR | Zbl

[11] Berenstein, Carlos A.; Gay, Roger; Vidras, Alekos; Yger, Alain Residue currents and Bezout identities, Progress in Mathematics, 114, Birkhäuser Verlag, Basel, 1993 | MR | Zbl

[12] Björk, Jan-Erik Residues and $𝒟$-modules, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 605-651 | MR

[13] Coleff, Nicolas R.; Herrera, Miguel E. Les courants résiduels associés à une forme méromorphe, Lecture Notes in Mathematics, 633, Springer, Berlin, 1978 | MR | Zbl

[14] Eisenbud, David Commutative algebra, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995 (With a view toward algebraic geometry) | MR | Zbl

[15] Henkin, Gennadi; Passare, Mikael Abelian differentials on singular varieties and variations on a theorem of Lie-Griffiths, Invent. Math., Volume 135 (1999) no. 2, pp. 297-328 | DOI | MR | Zbl

[16] Huneke, Craig Uniform bounds in Noetherian rings, Invent. Math., Volume 107 (1992) no. 1, pp. 203-223 | DOI | MR | Zbl

[17] Hyry, Eero; Villamayor U., Orlando A Briançon-Skoda theorem for isolated singularities, J. Algebra, Volume 204 (1998) no. 2, pp. 656-665 | DOI | MR | Zbl

[18] Jonsson, M.; Wulcan, E. On Bochner-Martinelli residue currents and their annihilator ideals, Ann. Inst. Fourier (Grenoble), Volume 59 (2009) no. 6, pp. 2119-2142 | DOI | Numdam

[19] Lejeune-Jalabert, M.; Teissier, B.; Risler, J.-J. Clôture intégrale des idéaux et équisingularité, Ann. Fac. Sci. Toulouse Math. (6), Volume 17 (2008) no. 4, pp. 781-859 | DOI | Numdam | MR | Zbl

[20] Lipman, Joseph; Sathaye, Avinash Jacobian ideals and a theorem of Briançon-Skoda, Michigan Math. J., Volume 28 (1981) no. 2, pp. 199-222 | DOI | MR | Zbl

[21] Lipman, Joseph; Teissier, Bernard Pseudorational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J., Volume 28 (1981) no. 1, pp. 97-116 | DOI | MR | Zbl

[22] Passare, Mikael; Tsikh, August; Yger, Alain Residue currents of the Bochner-Martinelli type, Publ. Mat., Volume 44 (2000) no. 1, pp. 85-117 | MR | Zbl

[23] Samuelsson, Håkan Regularizations of products of residue and principal value currents, J. Funct. Anal., Volume 239 (2006) no. 2, pp. 566-593 | DOI | MR | Zbl

[24] Scheja, Günter Riemannsche Hebbarkeitssätze für Cohomologieklassen, Math. Ann., Volume 144 (1961), pp. 345-360 | DOI | MR | Zbl

[25] Skoda, Henri Application des techniques $L^2$ à la théorie des idéaux d’une algèbre de fonctions holomorphes avec poids, Ann. Sci. École Norm. Sup. (4), Volume 5 (1972), pp. 545-579 | Numdam | MR | Zbl

[26] Skoda, Henri; Briançon, Joël Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de ${\mathbf{C}}^n$, C. R. Acad. Sci. Paris Sér. A, Volume 278 (1974), pp. 949-951 | MR | Zbl

[27] Sznajdman, J. An elementary proof of the Briançon-Skoda theorem (Preprint Gothenburg 2008, available at arXiv:0807.0142)

[28] Wulcan, Elizabeth Residue currents of monomial ideals, Indiana Univ. Math. J., Volume 56 (2007) no. 1, pp. 365-388 | DOI | MR | Zbl