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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2017
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1712.08112 |
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| _version_ | 1866910900540145664 |
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| author | Kelly, James P. Samuels, Charles L. |
| author_facet | Kelly, James P. Samuels, Charles L. |
| contents | 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 $\mathbb A_E$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1712_08112 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Direct Limits of Adèle Rings and Their Completions Kelly, James P. Samuels, Charles L. Number Theory 11R56, 13J10, 46A13 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 $\mathbb A_E$. |
| title | Direct Limits of Adèle Rings and Their Completions |
| topic | Number Theory 11R56, 13J10, 46A13 |
| url | https://arxiv.org/abs/1712.08112 |