Direct limits of adèle rings and their completions

Abstract

The adèle ring 𝔸K of a global field K is a locally compact, metrizable topological ring which is complete with respect to any invariant metric on 𝔸K. For a fixed global field F and a possibly infinite algebraic extension E∕F, there is a natural partial ordering on {𝔸K:F⊆K⊆E}. Therefore, we may form the direct limit 𝔸 E = lim → 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 𝕍¯E, to be a certain metrizable topological ring of continuous functions on the set of places of E. We show that 𝕍¯E is isomorphic to the completion of 𝔸E with respect to any invariant metric and use this isomorphism to establish several topological properties of 𝔸E.