Relative homological algebra and homological dimen- sion of Lie algebras

Abstract

Introduction. The main problem we shall consider concerns the various kinds of homological dimension that can be attached to a ring or to a pair consisting of a ring and a subring. We begin by investigating the functorial behavior of the relative Tor and Ext functors for pairs of rings R v S under a ring epimorphism R R/I. ?1 contains a general result describing this behavior. In ?2, this is applied to the case where R is the ordinary universal enveloping algebra of a restricted Lie algebra and R/I is the restricted universal enveloping algebra of that Lie algebra. We thus obtain an identification of the Tor and Ext functors of the restricted universal enveloping algebra with the relative Tor and Ext functors for the pair (R,S), where S is a subalgebra of R defined from the p-map of the restricted Lie algebra. The main purpose of ?3 is to prove a theorem on the coincidence of the several kinds of homological dimension for a restricted Lie algebra, which is precisely analogous to the well-known result of this type for ordinary Lie algebras. The proof is obtained from an appropriate adaptation of the requisite mapping theorem of Cartan-Eilenberg. ?4 gives an application of the general technique to the partial determination of the global homological dimension of certain factor algebras of the universal enveloping algebra of a Lie algebra, corresponding to a special subclass of the representations of the Lie algebra. These particular factor algebras of the universal enveloping algebra are of special interest, because as was shown by Sridharan in his Columbia thesis, they are precisely those algebras which possess a filtration such that the associated graded algebra is an ordinary polynomial algebra. In ?5, we show that the global homological dimension of the restricted universal enveloping algebra of a restricted Lie algebra is always either 0 or infinite. For a solvable restricted Lie algebra, we show that this global dimension is infinite if and only if the p-map has a nontrivial kernel. I would like to express here my deep indebtedness to Professor G. P. Hochschild for his generous help, kind advice, and guidance while this paper was in preparation. I am thankful to the referee for his helpful suggestions and improvements.