On noncommutative Noetherian schemes

Abstract

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 P.F. Smith on localization in Noetherian Artin-Rees rings. As an interesting corollary we obtain that every prime ideal of height 1 in the enveloping algebra of the Lie algebra sl(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.

Ore localization and minimal injective resolutions

“…Recall that a noetherian ring is an Artin-Rees (AR) ring if every prime ideal satisfies the Artin-Rees property (see [14,Chapter 13] for details on the Artin-Rees property in noncommutative ring theory). The class of noetherian AR rings in which every prime ideal is completely prime has good geometric properties (see, for example, [24]).…”

“…An interesting class of localizably correct prime ideals is provided by the work ofŠirola ([24]). By [32, Proposition 4.5 and Corollary 0.3], it follows that U(g), the universal enveloping algebra of a finite dimensional Lie algebra, satisfies the hypotheses in part b.)…”