THREE NOTES ON NICE EXTENSION DOMAINS

Abstract

The first note shows that the integral closure L′ of certain localities L over a local domain R are unmixed and analytically unramified, even when it is not assumed that R has these properties. The second note considers a separably generated extension domain B of a regular domain A, and a sufficient condition is given for a prime ideal p in A to be unramified with respect to B (that is, p B is an intersection of prime ideals and B/P is separably generated over A/p for all P ∈ Ass (B/p B)). Then, assuming that p satisfies this condition, a sufficient condition is given in order that all but finitely many q ∈ S = {q ∈ Spec(A), p ⊂ q and height(q/p) = 1} are unramified with respect to B, and a form of the converse is also considered. The third note shows that if R′ is the integral closure of a semi-local domain R, then I(R) = ∩{R′ p′ ;p′ ∈ Spec(R′) and altitude(R′/p′) = altitude(R′) − 1} is a quasi-semi-local Krull domain such that: (a) height(N *) = altitude(R) for each maximal ideal N * in I(R); and, (b) I(R) is an H-domain (that is, altitude(I(R)/p *) = altitude(I(R)) − 1 for all height one p * ∈ Spec(I(R))). Also, K = ∩{R p ; p ∈ Spec(R) and altitude(R/p) = altitude(R) − 1} is a quasi-semi-local H-domain such that height (N) = altitude(R) for all maximal ideals N in K.