Homological dimension of Ore-extensions

Global Dimension of Differential Operator Rings. II

“…As one consequence, we find that r gl dim R [0,,..., du] > u + rank(Ai) for any maximal ideal M of R such that R/M has positive characteristic. We begin with two lemmas, the first of which is essentially a special case of [6,Lemma,p. 68].…”

“…As a consequence, ker g is an R [6] -submodule of F, hence K can be made into a right R [8]-module so that /is an R [6]-homomorphism. Lemma 6.…”

“…Let R be a semiprime left Goldie differential ring. If J is any essential left ideal of R [6], then J contains an element of R [6] whose leading coefficient is a regular element of R.…”

“…Thus it suffices to consider only the 1-differential case: here i? is a semiprime right and left noetherian differential ring, A is a nonzero right R [6]-module such that AR is finitely generated, and we must prove that pdÄr0,(,4) = 1 + ^àR{A).…”

“…This element x can be put in the form x = x0 + xx(9 -a) for suitable x0GR and xx GR [9], whence t = x0(9 -a)~l + Xj. If x0 = 0, then / itself is left multiplication by the element x¡ GR [6] and the claim holds, hence we may assume that xQ =£ 0.…”