Decomposition of integer-valued polynomial algebras

Abstract

Let D be a commutative domain with field of fractions K, let A be a torsion-free D-algebra, and let B be the extension of A to a K-algebra. The set of integer-valued polynomials on A is Int(A)={f∈B[X]|f(A)⊆A}, and the intersection of Int(A) with K[X] is IntK(A), which is a commutative subring of K[X]. The set Int(A) may or may not be a ring, but it always has the structure of a left IntK(A)-module.A D-algebra A which is free as a D-module and of finite rank is called IntK-decomposable if a D-module basis for A is also an IntK(A)-module basis for Int(A); in other words, if Int(A) can be generated by IntK(A) and A. A classification of such algebras has been given when D is a Dedekind domain with finite residue rings. In the present article, we modify the definition of IntK-decomposable so that it can be applied to D-algebras that are not necessarily free by defining A to be IntK-decomposable when Int(A) is isomorphic to IntK(A)⊗DA. We then provide multiple characterizations of such algebras in the case where D is a discrete valuation ring or a Dedekind domain with finite residue rings. In particular, if D is the ring of integers of a number field K, we show that an IntK-decomposable algebra A must be a maximal D-order in a separable K-algebra B, whose simple components have as center the same finite unramified Galois extension F of K and are unramified at each finite place of F. Finally, when both D and A are rings of integers in number fields, we prove that IntK-decomposable algebras correspond to unramified Galois extensions of K.