On the Global Dimension of Certain Primitive Factors of the Enveloping Algebra of a Semi-Simple Lie Algebra

Abstract

Let g be a finite-dimensional semi-simple Lie algebra over an algebraically closed field k of characteristic zero and letg = n+rj + n~ be a triangular decomposition of g. Denote by U = U(Q) the enveloping algebra of g. For Xe\)*, denote by Dk the primitive factor ring £//t/ ^W> where M(X) is the Verma module of highest weight X—p (where p is the half-sum of the positive roots). The main aim of this paper is to prove Theorem 3.9, which states that if A is regular, then gldim Dx < dim . n + -In{X), where n{X) is a non-negative integer less than or equal to dimfcn+. In [12] Levasseur computed the injective dimension of D^ in terms of the Gelfand-Kirillov dimension of L{X), the unique simple quotient of M(X). For X regular, Theorem 3.9 implies that the global dimension of D^ is finite and hence must be equal to the injective dimension of Dk. It can be shown that this figure coincides with the bound given in Theorem 3.9 (the authors would like to thank Levasseur for pointing this out). On the other hand, if X is not regular, Joseph and Stafford [11] have shown that gldim D^= oo. Special cases of this result have already appeared in the literature. In particular Stafford [17] computed the global dimension of Dk for all A el)* in the case when g = si (2, C). On the other hand Roos [15] computed gldim D^ for general g when X satisfies some transcendental but generic conditions. This paper is divided into two distinct parts. In Section 2 we prove a result analogous to a theorem of Roos relating the weak global dimension of a ring R to that of certain sets of torsion-theoretic localisations. If R is a commutative ring and {Si)tei i a s e t °f localisations of R such that © S{ is faithfully flat as an /^-module, then it is well known that wgldim R = sup{wgldim St}. To what extent this result is true for non-commutative rings is unknown. Roos and others have proved various results under the assumption that wgldim R is finite (see for instance [3, 7, 13,14]). We prove here a slightly more general sort of result than Roos's which does not require this assumption. However, it is necessary to impose certain additional assumptions on the localisations St. The special case of this result needed for the second part is the following. Let R be a prime Noetherian ring and let St, i = 1, ..., n be a finite collection of rings lying between R and its quotient ring. If © St is a faithfully flat right /^-module and St ®R S} = S} ®R St as /?-/?-bimodules for all pairs (i,j), then