no. 1
Elementary construction of residue currents associated to Cohen–Macaulay ideals
[Construction élémentaire des courants résiduels associés aux idéaux Cohen–Macaulay]
Lärkäng, Richard1 ; Mazzilli, Emmanuel2
1 Chalmers University of Technology and the University of Gothenburg Department of Mathematics 412 96 Göteborg (Sweden)
2 Université Lille 1 Laboratoire Paul Painlevé U.M.R. CNRS 8524 U.F.R. de Mathématiques, cité scientifique F59 655 Villeneuve d’Ascq Cedex (France)
Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 377-391.

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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3164
Classification : 32A26, 32A27, 32C30, 32C37, 13C14
Keywords: residue currents, explicit construction, theory of integral representations, duality principle, Cohen–Macaulay ideals
Mot clés : courants résiduels, construction explicite, théorie des représentations intégrales, principe de dualité, idéaux Cohen–Macaulay

Lärkäng, Richard 1 ; Mazzilli, Emmanuel 2

1 Chalmers University of Technology and the University of Gothenburg Department of Mathematics 412 96 Göteborg (Sweden)
2 Université Lille 1 Laboratoire Paul Painlevé U.M.R. CNRS 8524 U.F.R. de Mathématiques, cité scientifique F59 655 Villeneuve d’Ascq Cedex (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2018__68_1_377_0,
     author = {L\"ark\"ang, Richard and Mazzilli, Emmanuel},
     title = {Elementary construction of residue currents associated to {Cohen{\textendash}Macaulay} ideals},
     journal = {Annales de l'Institut Fourier},
     pages = {377--391},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {1},
     year = {2018},
     doi = {10.5802/aif.3164},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3164/}
}
TY  - JOUR
AU  - Lärkäng, Richard
AU  - Mazzilli, Emmanuel
TI  - Elementary construction of residue currents associated to Cohen–Macaulay ideals
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 377
EP  - 391
VL  - 68
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3164/
DO  - 10.5802/aif.3164
LA  - en
ID  - AIF_2018__68_1_377_0
ER  - 
%0 Journal Article
%A Lärkäng, Richard
%A Mazzilli, Emmanuel
%T Elementary construction of residue currents associated to Cohen–Macaulay ideals
%J Annales de l'Institut Fourier
%D 2018
%P 377-391
%V 68
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3164/
%R 10.5802/aif.3164
%G en
%F AIF_2018__68_1_377_0
Lärkäng, Richard; Mazzilli, Emmanuel. Elementary construction of residue currents associated to Cohen–Macaulay ideals. Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 377-391. doi : 10.5802/aif.3164. https://aif.centre-mersenne.org/articles/10.5802/aif.3164/

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

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

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

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

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

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

[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.), Volume 107(149) (1978) no. 2, p. 163-174, 317 | Zbl

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

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

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

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

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

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

[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 ℂ n (2015) (http://arxiv.org/abs/1510.01965)

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

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

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

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

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

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

Cité par Sources :