Direct limits of adèle rings and their completions

Abstract

The adèle ring ${\mathbb{𝔸}}_K$ of a global field $K$ is a locally compact, metrizable topological ring which is complete with respect to any invariant metric on ${\mathbb{𝔸}}_K$. For a fixed global field $F$ and a possibly infinite algebraic extension $E∕F$, there is a natural partial ordering on $\left\{{\mathbb{𝔸}}_K:F\subseteq K\subseteq E\right\}$. Therefore, we may form the direct limit

$${\mathbb{𝔸}}_E=\underrightarrow{lim}{\mathbb{𝔸}}_K,$$

which provides one possible generalization of adèle rings to arbitrary algebraic extensions $E∕F$. In the case where $E∕F$ is Galois, we define an alternate generalization of the adèles, denoted by ${\overline{\mathbb{𝕍}}}_E$, to be a certain metrizable topological ring of continuous functions on the set of places of $E$. We show that ${\overline{\mathbb{𝕍}}}_E$ is isomorphic to the completion of ${\mathbb{𝔸}}_E$ with respect to any invariant metric and use this isomorphism to establish several topological properties of ${\mathbb{𝔸}}_E$.