Universal properties of integer-valued polynomial rings

摘要

Let D be an integral domain, and let A be a domain containing D with quotient field K. We will say that the extension A of D is poynomially complete if D is a polynomially dense subset of A, that is, if for all f is an element of K[X] with f (D) subset of A one has f (A) subset of A. We show that, for any set (X) under bar, the ring lnt(D-(X) under bar) of integervalued polynomials on D-(X) under bar is the free polynomially complete extension of D generated by X, provided only that D is not a finite field. We prove that a divisorial extension of a Krull domain D is polynomially complete if and only if it is unrainified, and has trivial residue field extensions, at the height one primes in D with finite residue field. We also examine, for any extension A of a domain D, the following three conditions: (a) A is a polynomially complete extension of D; (b) Int(A(n)) superset of Int(D-n) for every positive integer n; and (c) Int(A) superset of Int(D). In general one has (a) double right arrow (b) double right arrow (c). It is known that (a) double left right arrow (c) if D is a Dedekind domain. We prove various generalizations of this result, such as: (a) double left right arrow (c) if D is a Krull domain and A is a divisorial extension of D. Generally one has (b) double left right arrow (c) if the canonical D-algebra homomorphism phi(n) : circle times(n)(i=1) Int(D) -> Int (D-n) surjective for all positive integers n, where the tensor product is over D. Furthermore, phi(n) is an isomorphism for all it if D is a Krull domain such that Int(D) is flat as a D-module, or if D is a Prufer domain such that Int(D-m) = Int(D)(m) for every maximal ideal rn of D. (c) 2007 Elsevier Inc. All rights reserved.