On noncommutative Noetherian schemes

[摘要] The main aim of this paper is to better understand the localization technique for certain Noetherian rings like enveloping algebras of nilpotent Lie algebras. For such rings R we also give a conjectural definition of certain sheaves which should be affine objects naturally generalizing the classically defined structure sheaves in commutative theory. The corresponding sheaves associated to some R-modules might carry particularly interesting information; e.g., for representation theory of semisimple Lie groups. Next, we generalize one important theorem of RE Smith on localization in Noetherian Artin-Rees rings. As an interesting corollary we obtain that every prime ideal of height I in the enveloping algebra of the Lie algebra st(2) over a field of characteristic zero is localizable. Finally, we provide a number of concrete useful calculations for our main example, the enveloping algebra of the three-dimensional Heisenberg Lie algebra; and thus test both the proposed ideas and methods. In particular, we introduce the notion of a weakly normal element, generalizing the notion of a normal element. (C) 2004 Elsevier Inc. All rights reserved.

[发布日期] 2004-12-15

[关键词] Noetherian ring;ore set;localization;Artin-Rees property;Zariski topology;prime spectrum;classical structure sheaf;weakly normal element