Sur certains invariants des algèbres artiniennes commutatives

Alexandre Aleksandrov

doi:10.5802/crmath.589

Research article - Algebra, Algebraic geometry

Sur certains invariants des algèbres artiniennes commutatives
[On certain invariants of commutative artinian algebras]

Alexandre Aleksandrov ¹

¹ Institut du Contrôle Automatique de l’Académie des sciences de Russie, 65, rue Profsoyuznaya, GSP-7, Moscou, 117997, Fédération de Russie

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 751-759.

Abstract
Résumé

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 $ℂ$ 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 $τ ⩾ μ$ .

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 $ℂ$ 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 $τ ⩾ μ$ pour ses nombres de Tjurina et de Milnor.

Received: 2022-12-05
Revised: 2023-05-18
Accepted: 2023-11-10
Published online: 2024-09-17

DOI: 10.5802/crmath.589

Author's affiliations:

Alexandre Aleksandrov ¹

¹ Institut du Contrôle Automatique de l’Académie des sciences de Russie, 65, rue Profsoyuznaya, GSP-7, Moscou, 117997, Fédération de Russie

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     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},
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     doi = {10.5802/crmath.589},
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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/

References
Cited by

[1] A. G. Aleksandrov Deformations of bouquets of quasihomogeneous one-dimensional singularities, Funct. Anal. Appl., Volume 15 (1981), pp. 53-55 | DOI | Zbl

[2] A. G. Aleksandrov Duality and deformations of artinian algebras, Proceedings of the International Conference on the Theory of Rings, Algebras and Modules in Memory of A. I. Shirshov, Barnaul, USSR (1991), p. 134

[3] A. G. Aleksandrov Duality, derivations and deformations of zero-dimensional singularities, Zero-dimensional schemes. Proceedings of the International Conference held in Ravello, Italy, June 8-13, 1992, Walter de Gruyter (1994), pp. 11-31 | MR | Zbl

[4] Michael F. Atiyah; I. G. Macdonald Introduction to commutative algebra, Addison-Wesley Publishing Group, 1969 | MR | Zbl

[5] W. Bruns; J. Herzog Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, 1996 | Zbl

[6] N. Bourbaki Éléments de mathématique. Algèbre. Chapitres 1 à 3, Masson, 1970 | MR | Zbl

[7] T. H. Gulliksen On the length of faithful modules over artinian local rings, Math. Scand., Volume 31 (1972), pp. 78-82 | DOI | MR | Zbl

[8] J. Herzog; E. Kunz Der kanonische Module eines Cohen–Macaulay Rings, Lecture Notes in Mathematics, 238, Springer, 1971 | DOI

[9] A. A. Iarrobino Associated graded algebra of a Gorenstein Artin algebra, Memoirs of the American Mathematical Society, 514, American Mathematical Society, 1994 | DOI | Zbl

[10] E. Kunz Almost complete intersections are not Gorenstein rings, J. Algebra, Volume 28 (1974), pp. 111-115 | DOI | MR | Zbl

[11] E. Kunz Kähler Differentials, Advanced Lectures in Mathematics, Springer, 1986 | DOI | MR

[12] E. Looijenga; J. Steenbrink Milnor number and Tjurina number of a complete intersection, Math. Ann., Volume 271 (1985), pp. 121-124 | DOI | MR | Zbl

[13] V. P. Palamodov Cohomology of analytical algebras, Tr. Mosk. Mat. O-va, Volume 44 (1982), pp. 3-61 | MR | Zbl

[14] G. Scheja; U. Storch Über Spurfunktionen bei vollständigen Durchschnitten, J. Reine Angew. Math., Volume 278/279 (1975), pp. 174-190 | MR | Zbl

[15] G. Scheja; H. Wiebe Über Derivationen von lokalen analytischen Algebren, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971 & Convegno di Geometria, INDAM, Rome, 1972), Volume 1973, Academic Press Inc., 1973, pp. 161-192 | MR | Zbl

[16] W. Teter Rings which are a factor of a Gorenstein ring by its socle, Invent. Math., Volume 23 (1974), pp. 153-162 | DOI | MR | Zbl

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