Timeline for Categorical description of the restricted product (Adeles)
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 20, 2014 at 19:34 | comment | added | Urs Schreiber | The restricted product appears quite a bit in nature. The function field analogy in the guise of the (geometric) Langlands correspondence shows that the restricted product that gives the adeles is just a special case of the restricted product of function algebras on all formal discs of an arithmetic/algebraic curve, with the restriction being along the inclusion of the functions on the unpunctured disk. These restricted product function algebras appear naturally as parts of the Cech cocycles for bundles over curves. See section 3.2 of ncatlab.org/nlab/show/restricted+product#Frenkel05 | |
| May 9, 2012 at 18:40 | comment | added | paul garrett | Yes, I agree that "mapping-property characterizations", rather than "set-theoretic constructions" (=element-wise descriptions) are much better in many situations! Absolutely! The present example may not be the best illustration, but, yes, still, we can do this. And Neal Strickland's answer, that starts with $\hat{\mathbb Z}$ as the limit of $\mathbb Z/n$ (noting, as Neal S. does, that we need not factor $n$), shows another way that "the adele ring" simply appears, rather than needing to "be defined into existence". | |
| May 9, 2012 at 16:43 | comment | added | Konrad Voelkel | It might be true that the restricted product doesn't appear much more often than to define adelic reductive groups, but for me the solution of my problem was quite enlightening: Here is a way to convert an element-dependent description into an element-free one. That is a technique I will surely want to use in other contexts. Thank you for the Solenoids-description, I wasn't aware of that either (it seems I'm an adelic newbie). | |
| May 6, 2012 at 22:23 | history | edited | paul garrett | CC BY-SA 3.0 |
added 2 characters in body
|
| May 6, 2012 at 21:41 | history | answered | paul garrett | CC BY-SA 3.0 |