Timeline for Modules over Laurent series rings
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 25, 2011 at 22:30 | comment | added | JBorger | Here's a stab. Given a ring $R$ and an element $r\in R$, you can first complete $R$ with respect to the ideal $(r)$ and then invert the image of $r$ in the completion. Is that the kind of thing you want in question 1? | |
| Jul 25, 2011 at 22:27 | comment | added | JBorger | I guess I just don't understand what you really want in question 1. As I said, localization and completion are already general ring-theoretic constructions. And if you don't mind making choices, then what don't you like about first completing at a chosen ideal and then inverting a chosen multiplicative subset? | |
| Jul 25, 2011 at 18:49 | comment | added | Mike Shulman | Thanks for the correction; I clearly wasn't quite awake when I posted this. But why should the non-specialness of x=0 prevent there from being a positive answer to question 1? Shouldn't it just mean we'd have to make some choice in the construction? | |
| Jul 25, 2011 at 18:37 | history | edited | Mike Shulman | CC BY-SA 3.0 |
fixed error
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| Jul 25, 2011 at 13:29 | answer | added | S. Carnahan♦ | timeline score: 3 | |
| Jul 25, 2011 at 11:55 | comment | added | JBorger | Anyway, it seems unlikely that there will be a nice positive answer to question 1. From the geometric point of view, $k((x))$ is field of formal Laurent series at the point $x=0$ of the affine line. But this point is not special in any way. Indeed, the automorphism group of the line permutes the points transitively. The answer to the second question may very well be yes. You could ask that the $k((x))$-module be complete with respect to a topology defined by certain sub-$k[[x]]$-modules. You might want to look into Tate vector spaces, about which there is an MO question. | |
| Jul 25, 2011 at 11:27 | comment | added | JBorger | Mike, I'm a little confused. First, unless I'm misunderstanding you, the 2nd sentence of your 2nd paragraph is wrong. A $k[x]$-module on which the action of $x$ is invertible is a $k[x,x^{-1}]$-module. A $k(x)$-module is one on which the action of every nonzero element of $k[x]$ is invertible. (The first sentence is also a bit funny.) More to the point, in your question 1, localization and completion are already general ring-theoretic constructions. Is what you don't like about them the fact that you have to name the multiplicative set at which you localize and the ideal at which you complete? | |
| Jul 25, 2011 at 11:01 | answer | added | the L | timeline score: 3 | |
| Jul 25, 2011 at 9:28 | history | asked | Mike Shulman | CC BY-SA 3.0 |