Timeline for Categorical description of the restricted product (Adeles)
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 16, 2016 at 17:58 | history | edited | user9072 |
edited tags
|
|
| May 9, 2012 at 22:58 | answer | added | Buschi Sergio | timeline score: 2 | |
| May 9, 2012 at 18:02 | answer | added | Neil Strickland | timeline score: 18 | |
| May 9, 2012 at 16:37 | vote | accept | Konrad Voelkel | ||
| May 9, 2012 at 16:37 | comment | added | Konrad Voelkel | @KConrad: Thank you! That is what I was after. | |
| May 9, 2012 at 16:36 | history | edited | Konrad Voelkel | CC BY-SA 3.0 |
removed unnecessary text
|
| May 7, 2012 at 7:35 | comment | added | Martin Brandenburg | @KConrad: This is an answer. | |
| May 6, 2012 at 21:41 | answer | added | paul garrett | timeline score: 6 | |
| May 6, 2012 at 18:43 | comment | added | KConrad | There is a universal mapping property for adeles, due to Goldman and Sah (On a Special Class of Locally Compact Rings, J. Algebra 4 (1966), 71--95): if a locally compact ring $R$ is an algebra over a global field $K$, has no proper open ideal, and its closed maximal ideals have intersection $\{0\}$, then there is a unique $K$-algebra homomorphism ${\mathbf A}_K \rightarrow R$. So the adeles of $K$ are an initial object in a suitable category of locally compact $K$-algebras. | |
| May 6, 2012 at 18:07 | comment | added | KConrad | discrete and co-compact in $R$ (meaning $K$ is discrete in the subspace topology and $R/K$ is compact in the quotient topology), $R$ is not discrete or compact, and the intersection of the closed maximal ideals in $R$ is $\{0\}$, then $K$ is a global field and $R$ is the adele ring of $K$. | |
| May 6, 2012 at 18:06 | comment | added | KConrad | You say that the restricted product topology is not the subspace topology despite its name, and I think that means you are parsing the term incorrectly: think about it as (restricted product) topology rather than restricted (product topology), since the adeles themselves are an example of a (restricted product) of topological groups/rings. There is a beautiful characterization of the adele ring of a global field, due to Iwasawa (On the rings of valuation vectors, Annals of Math 57 (1953), 331--3356): if $R$ is a locally compact topological ring containing a subfield $K$ that is (contd.) | |
| May 6, 2012 at 15:23 | answer | added | Marc Palm | timeline score: 8 | |
| May 6, 2012 at 14:33 | history | edited | Konrad Voelkel | CC BY-SA 3.0 |
(added two more ideas)
|
| May 6, 2012 at 14:17 | history | asked | Konrad Voelkel | CC BY-SA 3.0 |