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Détermination de la dimension homologique globale des algèbres de Weyl. (French)
Stockholm University, Faculty of Science, Department of Mathematics. matematik.
1972 (English)In: Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B, ISSN 0151-0509, Vol. 274, no 1, A23-A26 p.Article in journal (Refereed) Published
##### Abstract [en]

(Review by Rinehart):Let An be the algebra obtained from a field K by adjoining variables x_1, · · · , x_n, y_1, · · · , y_n

subject to the relations x_iy_j −y_jx_i =\delta ij . If K has characteristic 0, the reviewer showed [Proc.

Amer. Math. Soc. 13 (1962), 341–346; MR0137747 (25 #1196)] that A_1 has global dimension 1.

For larger n there followed an inequality, but the problem of a precise determination has remained

open until now. The author shows that A_n has global dimension n in characteristic 0. His result is

independent of (and includes) the reviewer’s. His proof for general n is considerably simpler than

that of the reviewer for n = 1, although this fact may be obscured by his deduction of the result

from general considerations involving Gabriel’s localizations. The only localizations that arise in

this case involve passage from A_n to A_n \otimes

_{K[x_n]} K(x_n) (with the evident ring structure), and a

similar one for y_n. These rings are easily seen to have dimension at most n (using an inductive

hypothesis), and the author shows that it follows readily that the weak dimension of A_n is at most

n. This suffices, since An is Noetherian.

Reviewed by G. S. Rinehart

##### Place, publisher, year, edition, pages
1972. Vol. 274, no 1, A23-A26 p.
##### Keyword [en]
Weyl algebra, homological dimension
Mathematics
##### Identifiers
OAI: oai:DiVA.org:su-13326DiVA: diva2:179846
Available from: 2008-03-17 Created: 2008-03-17 Last updated: 2011-01-16Bibliographically approved

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##### By organisation
Department of Mathematics
Mathematics