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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


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.


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


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

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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