Elementary construction of residue currents associated to Cohen–Macaulay ideals
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, p. 377-391
For a Cohen–Macaulay ideal of holomorphic functions, we construct by elementary means residue currents whose annihilator is precisely the given ideal. We give two proofs that the currents have the prescribed annihilator, one using the theory of linkage, and another using an explicit division formula involving these residue currents to express the ideal membership.
Pour un idéal Cohen–Macaulay de fonctions holomorphes, nous construisons de manière élémentaire des courants résiduels qui s’annulent précisément sur cet idéal. Nous donnons deux constructions, l’une utilisant la théorie des idéaux en algèbre commutative, et l’autre utilisant des représentations intégrales qui donnent une décomposition dans l’idéal modulo ces courants résiduels.
Received : 2016-07-10
Revised : 2017-05-22
Accepted : 2017-06-26
Published online : 2018-04-18
DOI : https://doi.org/10.5802/aif.3164
Classification:  32A26,  32A27,  32C30,  32C37,  13C14
Keywords: residue currents, explicit construction, theory of integral representations, duality principle, Cohen–Macaulay ideals
@article{AIF_2018__68_1_377_0,
     author = {L\"ark\"ang, Richard and Mazzilli, Emmanuel},
     title = {Elementary construction of residue currents associated to Cohen--Macaulay ideals},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {1},
     year = {2018},
     pages = {377-391},
     doi = {10.5802/aif.3164},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_1_377_0}
}
Elementary construction of residue currents associated to Cohen–Macaulay ideals. Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 377-391. doi : 10.5802/aif.3164. https://aif.centre-mersenne.org/item/AIF_2018__68_1_377_0/

[1] Andersson, Mats Integral representation with weights. II. Division and interpolation, Math. Z., Tome 254 (2006) no. 2, pp. 315-332 | Article | Zbl 1104.32002

[2] Andersson, Mats; Wulcan, Elizabeth Residue currents with prescribed annihilator ideals, Ann. Sci. Éc. Norm. Supér., Tome 40 (2007) no. 6, pp. 985-1007 | Article | Zbl 1143.32003

[3] Berndtsson, Bo Weighted integral formulas, Several complex variables (Stockholm, 1987/1988), Princeton University Press (Mathematical Notes) Tome 38 (1993), pp. 160-187 | Zbl 0786.32003

[4] Berndtsson, Bo; Andersson, Mats Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier, Tome 32 (1982) no. 3, pp. 91-110 | Article | Zbl 0466.32001

[5] Coleff, Nicolas R.; Herrera, Miguel E. Les courants résiduels associés à une forme méromorphe, Springer, Lecture Notes in Mathematics, Tome 633 (1978), x+209 pages | Zbl 0371.32007

[6] Cox, David A.; Little, John; O’Shea, Donal Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra, Springer, Undergraduate Texts in Mathematics (2015), xvi+646 pages | Zbl 1335.13001

[7] Dautov, Š. A.; Henkin, Gennadi M. Zeros of holomorphic functions of finite order and weighted estimates for the solutions of the ¯-equation, Mat. Sb. (N.S.), Tome 107(149) (1978) no. 2, p. 163-174, 317 | Zbl 0421.32001

[8] Dickenstein, Alicia; Gay, Roger; Sessa, Carmen; Yger, Alain Analytic functionals annihilated by ideals, Manuscr. Math., Tome 90 (1996) no. 2, pp. 175-223 | Article | Zbl 0871.32009

[9] Dickenstein, Alicia; Sessa, Carmen Canonical representatives in moderate cohomology, Invent. Math., Tome 80 (1985) no. 3, pp. 417-434 | Article | Zbl 0556.32005

[10] Dolbeault, Pierre Courants résidus des formes semi-méromorphes, Séminaire Pierre Lelong (Analyse) (1970), Springer (Lecture Notes in Mathematics) Tome 205 (1971), pp. 56-70 | Zbl 0229.32005

[11] Eisenbud, David Commutative algebra, Springer, Graduate Texts in Mathematics, Tome 150 (1995), xvi+785 pages (With a view toward algebraic geometry) | Zbl 0819.13001

[12] Fouli, Louiza; Huneke, Craig What is a system of parameters?, Proc. Am. Math. Soc., Tome 139 (2011) no. 8, pp. 2681-2696 | Article | Zbl 1227.13002

[13] Herrera, Miguel E.; Lieberman, David I. Residues and principal values on complex spaces, Math. Ann., Tome 194 (1971), pp. 259-294 | Article | Zbl 0224.32012

[14] Lärkäng, Richard A comparison formula for residue currents (2012) (http://arxiv.org/abs/1207.1279 )

[15] Lärkäng, Richard Explicit versions of the local duality theorem in n (2015) (http://arxiv.org/abs/1510.01965 )

[16] Lärkäng, Richard; Samuelsson Kalm, Håkan Various approaches to products of residue currents, J. Funct. Anal., Tome 264 (2013) no. 1, pp. 118-138 | Article | Zbl 1271.32010

[17] Lundqvist, Johannes A local Grothendieck duality theorem for Cohen-Macaulay ideals, Math. Scand., Tome 111 (2012) no. 1, pp. 42-52 | Article | Zbl 1276.32002

[18] Mazzilli, Emmanuel Division des distributions et applications à l’étude d’idéaux de fonctions holomorphes, C. R., Math., Acad. Sci. Paris, Tome 338 (2004) no. 1, pp. 1-6 | Article | Zbl 1040.32007

[19] Mazzilli, Emmanuel Courants du type résiduel attachés à une intersection complète, J. Math. Anal. Appl., Tome 368 (2010) no. 1, pp. 169-177 | Article | Zbl 1218.32005

[20] Passare, Mikael Residues, currents, and their relation to ideals of holomorphic functions, Math. Scand., Tome 62 (1988) no. 1, pp. 75-152 | Article | Zbl 0633.32005

[21] Vasconcelos, Wolmer V. Computational methods in commutative algebra and algebraic geometry, Springer, Algorithms and Computation in Mathematics, Tome 2 (1998), xi+394 pages | Zbl 0896.13021