Localization, Catenarity and Dimensions in Rings with Lie Algebra Action

AbstractThis paper is devoted to the study of smash products R#U(g) where R is a Noetherian algebra and g is a finite-dimensional Lie algebra (usually nilpotent or solvable) acting as a derivation on R. The questions considered involve the prime ideals of both R and R#U(g), especially the height of the prime ideals and its connection to their g-height. This is applied to show that the ring R#U(g) is catenary in certain cases and to connect the height with the (Gelfand-Kirillov) dimension of the corresponding factor ring (seeking to generalize the fact that dim A = ht P + dim(A/P) when P is the prime ideal of a commutative affine algebra A). Another major theme is the study of homological properties and related concepts such as regularity; a typical result is that R is regular whenever the localization RP is regular for all g-invariant prime ideals P, provided that R is g-hypernormal