Direct Limits of Adèle Rings and Their Completions
James P. Kelly, Charles L. Samuels
TL;DR
This work generalizes adèle rings to arbitrary infinite Galois extensions by introducing a direct-limit object $\mathbb A_E = \varinjlim \mathbb A_K$ and a new completion $\overline{\mathbb A}_E$. It then constructs an alternate generalization $\overline{\mathbb V}_E$, a metrizable topological ring of continuous functions on the place set $Y_E$, and proves an isomorphism $\overline{\mathbb V}_E \cong \overline{\mathbb A}_E$, thereby realizing the completed adèles as a function space. The paper also develops a robust topology via $v$-adic transition diagrams and establishes that $\mathbb A_E$ is not complete in the infinite case, with $\mathbb V_E$ having empty interior in $\overline{\mathbb V}_E$, and discusses limitations such as potential non-local-compactness of the generalized objects. The results connect direct-limit adèles with a functional-analytic viewpoint, enabling topological and completion analyses for infinite Galois extensions and clarifying the role of conorms in the structure.
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 $\{\mathbb A_K:F\subseteq K\subseteq E\}$. Therefore, we may form the direct limit \[ \mathbb A_E = \varinjlim \mathbb A_K \] 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 $\bar{\mathbb V}_E$, to be a certain metrizable topological ring of continuous functions on the set of places of $E$. We show that $\bar{\mathbb V}_E$ 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 $\mathbb A_E$.