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June 2020 Direct limits of adèle rings and their completions
James P. Kelly, Charles L. Samuels
Rocky Mountain J. Math. 50(3): 1021-1043 (June 2020). DOI: 10.1216/rmj.2020.50.1021

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 EF, there is a natural partial ordering on {𝔸K:FKE}. Therefore, we may form the direct limit

𝔸 E = lim 𝔸 K ,

which provides one possible generalization of adèle rings to arbitrary algebraic extensions EF. In the case where EF 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.

Citation

Download Citation

James P. Kelly. Charles L. Samuels. "Direct limits of adèle rings and their completions." Rocky Mountain J. Math. 50 (3) 1021 - 1043, June 2020. https://doi.org/10.1216/rmj.2020.50.1021

Information

Received: 14 August 2019; Revised: 4 December 2019; Accepted: 19 December 2019; Published: June 2020
First available in Project Euclid: 29 July 2020

zbMATH: 07235594
MathSciNet: MR4132624
Digital Object Identifier: 10.1216/rmj.2020.50.1021

Subjects:
Primary: 11R56, 13J10, 46A13
Secondary: 11R32, 18A30, 22D15, 54A20

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.50 • No. 3 • June 2020
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