Détermination de la dimension homologique globale des algèbres de Weyl. (French)
Détermination de la dimension homologique globale des algèbres de Weyl. (French)
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（法语）
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发表时间：
1972
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通讯作者：
Jan
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作者：
Jan
（由莱因哈特复习）：设An是由域K由相邻变量x_1，···，x_n, y_1，···，y_n根据x_iy_j−y_jx_i = ij得到的代数。如果K具有特征0，审稿人显示[procc . amer]。数学。《社会法学》13 (1962),341-346；MR0137747 (25 #1196)]， A_1具有全局维数1。对于更大的n，后面有一个不等式，但精确确定的问题直到现在还没有解决。证明了A_n在特征0上具有全局维数n。他的结果独立于（也包括）审稿人的结果。他对一般n的证明比审稿人对n = 1的证明要简单得多，尽管这一事实可能被他从涉及加布里埃尔局域化的一般考虑中推导出来的结果所掩盖。在这种情况下出现的唯一本地化涉及从A_n到A_n的传递__{K[x_n]} K(x_n)（具有明显的环结构），以及y_n的类似本地化。这些环的维数最多为n（使用归纳假设），作者很容易得出A_n的弱维数最多为n的结论。这就足够了，因为An是诺etherian。G. S.莱因哈特评论
(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