Localisation in rings equipped with a solvable Lie algebra action

Abstract

Let k be an algebraically closed uncountable field of characteristic 0,g a finite dimensional solvable k-Lie algebraR a noetherian k-algebra on which g acts by k-derivationsU(g) the enveloping algebra of g,A=R*g the crossed product of R by U(g)P a prime ideal of A and Ω(P) the clique of P. Suppose that the prime ideals of the polynomial ring R[x] are completely prime. If R is g-hypernormal, then Ω(P) is classical. Denote by AT the localised ring and let M be a primitive ideal of AT Set Q=P∩R In this note, we show that if R is a strongly (R,g)-admissible integral domain and if QRQ is generated by a regular g-centralising set of elements, then (1)M is generated by a regular g-semi-invariant normalising set of elements of cardinald = dim (RQ 0 + ∣XA (P)∣ (2)d gldim(AT ) = Kdim(AT ) = ht(M) = ht(P).