Homological properties of noncommutative Iwasawa algebras

Wei, Feng

doi:10.1016/j.crma.2010.11.030

Algebra/Homological Algebra

Homological properties of noncommutative Iwasawa algebras
[Propriétés homologiques des algèbres d'Iwasawa non commutatives]

Présenté par : Christophe Soulé

Wei, Feng ¹

¹ Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, China

Comptes Rendus. Mathématique, Tome 349 (2011) no. 1-2, pp. 15-20.

Résumé
Abstract

Pour tout groupe de Lie G p-adique compact, l'algèbre d'Iwasawa $Λ_{G}$ et son image épimorphique $Ω_{G}$ sont des algèbres d'Artin–Schelter Gorenstein. Nous montrons la formule d'Auslander–Buchsbaum, le théorème de Bass et le théorème des « non trous » pour des modules noethériens sur $Ω_{G}$ , ainsi que des versions duales pour leur modules artiniens. Il est montré que $Ω_{G}$ est auto-duale au sens de Morita par des complexes dualisants. Finalement, nous considérons les invariants homologiques « grade » des modules filtrés sur $Λ_{G}$ et $Ω_{G}$ , lorsque G est un groupe uniforme pro-p satisfaisant certaines propriétés.

For any compact p-adic Lie group G, the Iwasawa algebra $Ω_{G}$ is an Artin–Schelter Gorenstein algebra. We obtain the Auslander–Buchsbaum formula, the Bass's theorem and the No-holes theorem for noetherian modules over $Λ_{G}$ and $Ω_{G}$ , and the dual versions for their artinian modules. It is shown that $Ω_{G}$ is Morita self-dual via dualizing complexes. We finally consider the homological invariant “grade” of filtered modules over $Λ_{G}$ and $Ω_{G}$ , when G is a uniform pro-p group with certain properties.

Reçu le : 2010-09-01
Accepté le : 2010-11-29
Publié le : 2010-12-18

DOI : 10.1016/j.crma.2010.11.030

Affiliations des auteurs :

Wei, Feng ¹

¹ Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, China

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Wei, Feng. Homological properties of noncommutative Iwasawa algebras. Comptes Rendus. Mathématique, Tome 349 (2011) no. 1-2, pp. 15-20. doi : 10.1016/j.crma.2010.11.030. http://www.numdam.org/articles/10.1016/j.crma.2010.11.030/

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