Open Access Open Access  Restricted Access Subscription Access

A global Briançon-Skoda-Huneke-Sznajdman theorem

Mats Andersson


We prove a global effective membership result for polynomials on a non-reduced algebraic subvariety of $\mathbb{C}^N$. It can be seen as a global version of a recent local result of Sznajdman, generalizing the Briançon-Skoda-Huneke theorem for the local ring of holomorphic functions at a point on a reduced analytic space.

Full Text:



Andersson, M., Residue currents and ideals of holomorphic functions, Bull. Sci. Math. 128 (2004), no. 6, 481–512.

Andersson, M., Uniqueness and factorization of Coleff-Herrera currents, Ann. Fac. Sci. Toulouse Math. (6) 18 (2009), no. 4, 651–661.

Andersson, M. and Samuelsson, H., A Dolbeault-Grothendieck lemma on complex spaces via Koppelman formulas, Invent. Math. 190 (2012), no. 2, 261–297.

Andersson, M., Samuelsson, H., and Sznajdman, J., On the Briançon-Skoda theorem on a singular variety, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 2, 417–432.

Andersson, M. and Wulcan, E., Residue currents with prescribed annihilator ideals, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 6, 985–1007.

Andersson, M. and Wulcan, E., Decomposition of residue currents, J. Reine Angew. Math. 638 (2010), 103–118.

Andersson, M. and Wulcan, E., Direct images of semi-meromorphic currents, preprint arXiv:1411.4832 [math.CV], 2014.

Andersson, M. and Wulcan, E., Global effective versions of the Briançon-Skoda-Huneke theorem, Invent. Math. 200 (2015), no. 2, 607–651.

Björk, J.-E., Rings of differential operators, North-Holland Mathematical Library, vol. 21, North-Holland Publishing Co., Amsterdam-New York, 1979.

Björk, J.-E., Residues and $\mathcal{D}$-modules, in “The legacy of Niels Henrik Abel”, Springer, Berlin, 2004, pp. 605--651.

Demailly, J.-P., Complex analytic and differential geometry, Monograph, Grenoble, 1997.

Ehrenpreis, L., Fourier analysis in several complex variables, Pure and Applied Mathematics, Vol. XVII, Wiley-Interscience Publishers, 1970.

Ein, L. and Lazarsfeld, R., A geometric effective Nullstellensatz, Invent. Math. 137 (1999), no. 2, 427–448.

Eisenbud, D., Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995.

Eisenbud, D., The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005.

Herrera, M. and Lieberman, D., Residues and principal values on complex spaces, Math. Ann. 194 (1971), 259–294.

Hickel, M., Solution d'une conjecture de C. Berenstein-A. Yger et invariants de contact à l'infini, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 3, 707–744.

Huneke, C., Uniform bounds in Noetherian rings, Invent. Math. 107 (1992), no. 1, 203–223.

Jelonek, Z., On the effective Nullstellensatz, Invent. Math. 162 (2005), no. 1, 1–17.

Kollár, J., Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (1988), no. 4, 963–975.

Lipman, J. and Sathaye, A., Jacobian ideals and a theorem of Briançon-Skoda, Michigan Math. J. 28 (1981), no. 2, 199–222.

Lipman, J. and Teissier, B., Pseudorational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), no. 1, 97–116.

Oberst, U., The construction of Noetherian operators, J. Algebra 222 (1999), no. 2, 595–620.

Palamodov, V. P., Linear differential operators with constant coefficients, Die Grundlehren der mathematischen Wissenschaften, Band 168, Springer-Verlag, New York-Berlin, 1970.

Passare, M., Tsikh, A., and Yger, A., Residue currents of the Bochner-Martinelli type, Publ. Mat. 44 (2000), no. 1, 85–117.

Skoda, H. and Briançon, J., 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 278 (1974), 949–951.

Sznajdman, J., A Briançon-Skoda type result for a non-reduced analytic space, J. Reine Angew. Math. (2016), online ahead of print.

Vidras, A. and Yger, A., Briançon-Skoda theorem for a quotient ring, in “Complex analysis and dynamical systems VI. Part 2”, Contemp. Math., vol. 667, Amer. Math. Soc., Providence, RI, 2016, pp. 253--278.



  • There are currently no refbacks.
This website uses cookies to allow us to see how the site is used. The cookies cannot identify you or any content at your own computer.

ISSN 0025-5521 (print) ISSN 1903-1807 (online)

Hosted by the Royal Danish Library