Direct Limits of Adèle Rings and Their Completions

• Published in: arXiv.org 2019-12
• Main Authors: Kelly, James P, Samuels, Charles L
• Format: Article
• Language: English
• Subjects: Continuity (mathematics); Rings (mathematics); Topology
• ISSN: 2331-8422

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 \(\mathbb A_E\).