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.
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.
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Michał Lasoń 1, 2
@article{CRMATH_2012__350_15-16_737_0, author = {Micha{\l} Laso\'n}, title = {Equivalent condition for approximately {Cohen{\textendash}Macaulay} complexes}, journal = {Comptes Rendus. Math\'ematique}, pages = {737--739}, publisher = {Elsevier}, volume = {350}, number = {15-16}, year = {2012}, doi = {10.1016/j.crma.2012.09.004}, language = {en}, }
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/
[1] On sequentially Cohen–Macaulay complexes and posets, Israel J. Math., Volume 169 (2009), pp. 295-316
[2] On sequentially Cohen–Macaulay modules | arXiv
[3] Resolutions of Stanley–Reisner rings and Alexander duality, J. Pure Appl. Algebra, Volume 130 (1989), pp. 265-275
[4] Approximately Cohen–Macaulay rings, J. Algebra, Volume 76 (1982) no. 1, pp. 214-225
[5] Componentwise linear ideals, Nagoya Math. J., Volume 153 (1999), pp. 141-153
[6] On the full, strongly exceptional collections on toric varieties with Picard number three, Collect. Math., Volume 62 (2011) no. 3, pp. 275-296
[7] Cohen–Macaulay quotients of polynomial rings, Adv. Math., Volume 21 (1976) no. 1, pp. 30-49
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