Direct Limits of Adèle Rings and Their Completions

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...

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Bibliographic Details
Published in:arXiv.org 2019-12
Main Authors: Kelly, James P, Samuels, Charles L
Format: Article
Language:English
Subjects:
Continuity (mathematics)
Rings (mathematics)
Topology
Online Access:Get full text
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Summary: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 \(\mathbb A_E\).
ISSN:2331-8422