Exercise 27, § 4

Let A be an integrally closed noetherian local ring. Suppose moreover that A is a Nagata ring. Let x be an element of the field of fractions K of A, not belonging to A. Let B be the ring A[x] and p a maximal ideal of B containing x. Put I = xA \cap A. a) Let X be an indeterminate and Q the kernel of the homomorphism of A[X] into B which maps X to x. Then Q is generated by the polynomials of the form aX − b, where a and b are elements of A such that ax = b, and I = xA \cap A is generated by the constant terms b of these polynomials. Moreover the inclusion of A in B induces an isomorphism of A/I onto B/xB. b) The ideal I of A is divisorial. c) \operatorname{Ass}_B(B/xB) contains no immersed prime ideal. d) Let q be a prime ideal of B_p associated with B_p/xB_p. Then q \cap A is associated with A/I and the rings A_{q \cap A} and (B_p)_q are discrete valuation rings. e) B_p/q is a Nagata ring. (Remark that B/(q \cap B) is isomorphic to A/(q \cap A).) f) The completion of B_p/q is reduced. g) The completion of B_p is reduced and the integral closure of B_p is finite over B_p. (Use Exercise 24.)