Universal properties of integer-valued polynomial rings

Abstract

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 polynomially complete if D is a polynomially dense subset of A, that is, if for all f ∈ K [ X ] with f ( D ) ⊆ A one has f ( A ) ⊆ A . We show that, for any set X ̲ , the ring Int ( D X ̲ ) of integer-valued polynomials on D X ̲ 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 unramified, 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 ) ⊇ Int ( D n ) for every positive integer n; and (c) Int ( A ) ⊇ Int ( D ) . In general one has (a) ⇒ (b) ⇒ (c). It is known that (a) ⇔ (c) if D is a Dedekind domain. We prove various generalizations of this result, such as: (a) ⇔ (c) if D is a Krull domain and A is a divisorial extension of D. Generally one has (b) ⇔ (c) if the canonical D-algebra homomorphism φ n : ⊗ i = 1 n Int ( D ) → Int ( D n ) is surjective for all positive integers n, where the tensor product is over D. Furthermore, φ n is an isomorphism for all n if D is a Krull domain such that Int ( D ) is flat as a D-module, or if D is a Prüfer domain such that Int ( D m ) = Int ( D ) m for every maximal ideal m of D.