no. 1
Algebra
On the number of generators of an algebra
[Sur le nombre de générateurs d'une algèbre]

Présenté par : Jean-Pierre Serre

First, Uriya A.1 ; Reichstein, Zinovy1
1 Department of Mathematics, University of British Columbia, Vancouver, Canada
Comptes Rendus. Mathématique, Tome 355 (2017) no. 1, pp. 5-9.

Un théorème classique de Forster affirme que tout module M de type fini et de rang ≤n sur un anneau noethérien de dimension de Krull d peut être engendré par n+d éléments. Nous prouvons une généralisation de ce résultat où le mot « module » est remplacé par « algèbre ». Les algèbres considérées ici sont de type fini, mais non nécessairement unitaires, commutatives ou même associatives. Le théorème de Forster peut être déduit du cas particulier où un module est vu comme une algèbre dont le produit de deux éléments quelconques est 0.

A classical theorem of Forster asserts that a finite module M of rank ≤n over a Noetherian ring of Krull dimension d can be generated by n+d elements. We prove a generalization of this result, with “module” replaced by “algebra”. Here we allow arbitrary finite algebras, not necessarily unital, commutative or associative. Forster's theorem can be recovered as a special case by viewing a module as an algebra where the product of any two elements is 0.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.11.015
First, Uriya A. 1 ; Reichstein, Zinovy 1

1 Department of Mathematics, University of British Columbia, Vancouver, Canada 
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First, Uriya A.; Reichstein, Zinovy. On the number of generators of an algebra. Comptes Rendus. Mathématique, Tome 355 (2017) no. 1, pp. 5-9. doi : 10.1016/j.crma.2016.11.015. http://www.numdam.org/articles/10.1016/j.crma.2016.11.015/

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