Dans cette note, on étudie les propriétés des invariants algébriques, topologiques et analytiques des algèbres artiniennes commutatives et les relations remarquables entre eux. Par exemple, on montre que la longueur du module des différentielles de Kähler de toute algèbre artinienne locale de Gorenstein sur est supérieure ou égale à la longueur de l’algèbre lui-mème moins un. On obtient alors, grâce à la dualité canonique dans le complexe cotangent, que si le point épais correspondant est lissifiable, on a l’inégalité pour ses nombres de Tjurina et de Milnor.
We study properties of algebraic, topological and analytic invariants of commutative artinian algebras and relationships between them. For example, we show that the length of the module of Kähler differentials of any local artinian Gorenstein algebra over is greater than or equal to the length of the algebra itself minus one. We then prove, employing the canonical duality in the cotangent complex, that if the corresponding thick point is smoothable, then its Tjurina and Milnor numbers satisfy the inequality .
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Alexandre Aleksandrov 1
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@article{CRMATH_2024__362_G7_751_0,
author = {Alexandre Aleksandrov},
title = {Sur certains invariants des alg\`ebres artiniennes commutatives},
journal = {Comptes Rendus. Math\'ematique},
pages = {751--759},
publisher = {Acad\'emie des sciences, Paris},
volume = {362},
year = {2024},
doi = {10.5802/crmath.589},
language = {fr},
}
Alexandre Aleksandrov. Sur certains invariants des algèbres artiniennes commutatives. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 751-759. doi : 10.5802/crmath.589. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.589/
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