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- Roos, Jan-Erik
(author)
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Détermination de la dimension homologique globale des algèbres de Weyl. (French)
- 1972
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In: Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B. - 0151-0509. ; 274:1, s. A23-A26
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Journal article (peer-reviewed)abstract
- (Review by Rinehart):Let An be the algebra obtained from a field K by adjoining variables x_1, · · · , x_n, y_1, · · · , y_nsubject 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 remainedopen until now. The author shows that A_n has global dimension n in characteristic 0. His result isindependent of (and includes) the reviewer’s. His proof for general n is considerably simpler thanthat of the reviewer for n = 1, although this fact may be obscured by his deduction of the resultfrom general considerations involving Gabriel’s localizations. The only localizations that arise inthis case involve passage from A_n to A_n \otimes_{K[x_n]} K(x_n) (with the evident ring structure), and asimilar one for y_n. These rings are easily seen to have dimension at most n (using an inductivehypothesis), and the author shows that it follows readily that the weak dimension of A_n is at mostn. This suffices, since An is Noetherian.Reviewed by G. S. Rinehart
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