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 $\{{\mathbb{A}}_K:F\subseteq K\subseteq E\}$. 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 ${\mathbb{A}}_E$.

18 pages

References in corpus (1)

• A Banach space determined by the Weil height

Cited by in corpus (2)

• Rigid inner forms over local function fields
• A classification of $\mathbb{Q}$-valued linear functionals on ${\overline{\mathbb{Q}}}^{\times }$ modulo units