Tauvel’s Height Formula in Iterated Differential Operator Rings

Let k be a field of positive characteristic, R an associative algebra over k and let ∆1,n = {δ1, . . . , δn} be a finite set of k-linear derivations from R to R. Let A = Rn = R[θ1, δ1] · · · [θn, δn] be an iterated differential operator k-algebra over R such that δj(θi) ∈ Ri−1θi +Ri−1; 1 ≤ i < j ≤ n. As central result we show that if R is noetherian affine ∆1,n-hypernormal and if Tauvel’s height formula holds for the ∆1,n-prime ideals of R, then Tauvel’s height formula holds in A. In particular, let g be a completely solvable finite-dimensional k-Lie algebra acting by derivations on R and let U(g) be the enveloping algebra of g. If R is noetherian affine g-hypernormal and if Tauvel’s height formula holds for the g-prime ideals of R, then Tauvel’s height formula holds in the crossed product of R by U(g).