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.


Andersson, M., Residue currents and ideals of holomorphic functions, Bull. Sci. Math. 128 (2004), no. 6, 481–512. https://doi.org/10.1016/j.bulsci.2004.03.003

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. https://doi.org/10.1007/s00222-012-0380-9

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. https://doi.org/10.1016/j.ansens.2007.11.001

Andersson, M. and Wulcan, E., Decomposition of residue currents, J. Reine Angew. Math. 638 (2010), 103–118. https://doi.org/10.1515/CRELLE.2010.004

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. https://doi.org/10.1007/s00222-014-0544-x

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. https://doi.org/10.1007/s002220050332

Eisenbud, D., Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. https://doi.org/10.1007/978-1-4612-5350-1

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. https://doi.org/10.1007/BF01350129

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. https://doi.org/10.1007/BF01231887

Jelonek, Z., On the effective Nullstellensatz, Invent. Math. 162 (2005), no. 1, 1–17. https://doi.org/10.1007/s00222-004-0434-8

Kollár, J., Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (1988), no. 4, 963–975. https://doi.org/10.2307/1990996

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. https://doi.org/10.1006/jabr.1999.8035

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. https://doi.org/10.5565/PUBLMAT_44100_02

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. https://doi.org/https://doi.org/10.1515/crelle-2015-0099

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. https://doi.org/10.1090/conm/667/13545




How to Cite

Andersson, M. (2018). A global Briançon-Skoda-Huneke-Sznajdman theorem. MATHEMATICA SCANDINAVICA, 122(1), 31–52. https://doi.org/10.7146/math.scand.a-97253