Timeline for Localization of module
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Jun 29, 2010 at 1:13 | comment | added | KConrad | Localization solves a very specific universal mapping problem, so if you want to respect that feature of localization then the answer to your question will be no: all possible means of creating a module structure will lead to isomorphic modules. It's like asking if the integers can be given two essentially different fraction fields (it can't). | |
| Jun 29, 2010 at 1:11 | comment | added | KConrad | Maybe you want to clarify what it is you really want to see. For example, in the "exotic" S^{-1}A-module structure do you want to insist that the A-scaling is the same as it was originally? I suspect you meant to ask for that (which would nullify my previous example), so please write out your question a little more carefully. | |
| Jun 29, 2010 at 1:08 | comment | added | KConrad | Kwan, are you looking for an example or do you really need to keep all the data (M, A, S) completely general? You're basically asking if you can take a module over a ring and give the underlying additive group of the module another module structure over that ring which is not isomorphic to the first module structure. There is a standard instance of this. Let F be a field and V be a finite-dim. F-vector space. If dim(V) > 1, we can give V lots of nonisomorphic F[x]-module structures: this is the theme of Jordan canonical form (or rational canonical form if F isn't algebraically closed). | |
| Jun 29, 2010 at 0:54 | comment | added | ashpool | @Harry Altman It's a long story. | |
| Jun 29, 2010 at 0:51 | comment | added | Harry Altman | My first inclination on seeing this is to ask, why would you want such a thing? | |
| Jun 29, 2010 at 0:47 | history | edited | ashpool | CC BY-SA 2.5 |
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| Jun 29, 2010 at 0:40 | history | asked | ashpool | CC BY-SA 2.5 |