Equivalent condition for approximately Cohen–Macaulay complexes

Michał Lasoń

doi:10.1016/j.crma.2012.09.004

Algebra

Equivalent condition for approximately Cohen–Macaulay complexes
[Condition équivalente pour les complexes approximativement Cohen–Macaulay]

Présenté par : the Editorial Board

Michał Lasoń ^1,²

¹ Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland
² Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland

Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 737-739.

Résumé
Abstract

Nous donnons une condition nécessaire et suffisante pour quʼun complexe simplicial soit approximativement Cohen–Macaulay. Précisément, un complexe est approximativement Cohen–Macaulay si et seulement si lʼidéal associé à son dual dʼAlexander est engendré en deux degrés consécutifs et chacune de ses composantes a une résolution linéaire. Cela complète le résultat de J. Herzog et T. Hibi, qui démontrent quʼun complexe simplicial est séquentiellement Cohen–Macaulay si et seulement si chacune des composantes de lʼidéal associé à son dual dʼAlexander a une résolution linéaire.

We give a necessary and sufficient condition for a simplicial complex to be approximately Cohen–Macaulay. Namely it is approximately Cohen–Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear and generated in two consecutive degrees. This completes the result of J. Herzog and T. Hibi who proved that a simplicial complex is sequentially Cohen–Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear.

Reçu le : 2012-05-07
Accepté le : 2012-09-06
Publié le : 2012-08-01

DOI : 10.1016/j.crma.2012.09.004

Affiliations des auteurs :

Michał Lasoń ^{1, 2}

¹ Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland
² Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland

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     title = {Equivalent condition for approximately {Cohen{\textendash}Macaulay} complexes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {737--739},
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     doi = {10.1016/j.crma.2012.09.004},
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Michał Lasoń. Equivalent condition for approximately Cohen–Macaulay complexes. Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 737-739. doi : 10.1016/j.crma.2012.09.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.09.004/

Bibliographie
Cité par

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[2] N. Cuong; D. Cuong On sequentially Cohen–Macaulay modules | arXiv

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[4] S. Goto Approximately Cohen–Macaulay rings, J. Algebra, Volume 76 (1982) no. 1, pp. 214-225

[5] J. Herzog; T. Hibi Componentwise linear ideals, Nagoya Math. J., Volume 153 (1999), pp. 141-153

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