Decomposition of integer-valued polynomial algebras

TL;DR

This paper studies the structure and classification of certain integer-valued polynomial algebras over number rings, focusing on their decomposition properties and connections to algebraic number theory.

Contribution

It introduces a generalized definition of ${ m Int}_K$-decomposable algebras applicable to non-free modules and characterizes these algebras over Dedekind domains and number rings.

Findings

Characterization of ${ m Int}_K$-decomposable algebras over Dedekind domains.

Connection between ${ m Int}_K$-decomposable algebras and maximal orders in separable algebras.

Correspondence of such algebras to unramified Galois extensions of number fields.

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 ${\rm Int}(A) = \{f \in B[X] \mid f(A) \subseteq A\}$, and the intersection of ${\rm Int}(A)$ with $K[X]$ is ${\rm Int}_K(A)$, which is a commutative subring of $K[X]$. The set ${\rm Int}(A)$ may or may not be a ring, but it always has the structure of a left ${\rm Int}_K(A)$-module. A $D$-algebra $A$ which is free as a $D$-module and of finite rank is called ${\rm Int}_K$-decomposable if a $D$-module basis for $A$ is also an ${\rm Int}_K(A)$-module basis for ${\rm Int}(A)$; in other words, if ${\rm Int}(A)$ can be generated by ${\rm Int}_K(A)$ 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 ${\rm Int}_K$-decomposable so that it can be applied to $D$-algebras that are not necessarily free by defining $A$ to be ${\rm Int}_K$-decomposable when ${\rm Int}(A) \cong {\rm Int}_K(A) \otimes_D A$. 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 ${\rm Int}_K$-decomposable algebras $A$ correspond to maximal $D$-orders 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 show that ${\rm Int}_K$-decomposable algebras correspond to unramified Galois extensions of $K$.