Timeline for Topologists loops versus algebraists loops
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
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| Feb 26, 2010 at 14:20 | comment | added | Anatoly Preygel | Vague as I was, I realize I was wrong on how to topologize it (and this might help?). It should be topologized as with the subspace topology (viewed as pairs $(b,b^{-1})$) inside $\mathbb{C}((t))^2$, where this latter is (I think) given the topology I was describing. I can't tell if this helps, but it might. | |
| Feb 26, 2010 at 14:04 | comment | added | Anatoly Preygel | The algebraic loop space is not the base change but the restriction of scalars from $\mathbb{C}$ to $\mathbb{C}((t))$. So, its $\mathbb{C}$ points are by definition $\mathbb{G}_m(\mathbb{C}((t))=\mathbb{C}((t))^{\times}$: the non-zero Laurent series. As to how we topologize it, my guess is as a direct limit of product topologies. Both $L \mathbb{G}_m$ and this have a subspace homeomorphic to $\mathbb{Z} \times \mathbb{G}_m$: loops of the form $z \mapsto t z^n$ for some $(n,t)$. The inclusion is an equiv. for the topological one, what about the alg. geom. one? | |
| Feb 26, 2010 at 13:47 | history | edited | Chris Schommer-Pries | CC BY-SA 2.5 |
Added a few more insights.
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| Feb 26, 2010 at 13:19 | history | answered | Chris Schommer-Pries | CC BY-SA 2.5 |