Timeline for Infinite-dimensional affine space in algebraic geometry and algebraic topology
Current License: CC BY-SA 4.0
14 events
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| S 6 hours ago | history | suggested | Smiley1000 | CC BY-SA 4.0 |
Improve formatting: "$Spec$" -> "$\operatorname{Spec}$"; "$Proj$" -> "$\operatorname{Proj}$"
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| 13 hours ago | review | Suggested edits | |||
| S 6 hours ago | |||||
| Feb 10, 2021 at 22:15 | comment | added | Will Sawin | There are infinite-dimensional compact vectors spaces in ordinary topology, too (over finite fields). Other words, I concur entirely with Sam Gunningham - both come up a lot in geometric Langlands and related fields, and I haven't seen any other examples (unless you count the product of those two as an example). | |
| Feb 8, 2021 at 22:44 | comment | added | Sam Gunningham | The ind-scheme $\mathbb A^\infty$ and the infinite-type scheme $\mathbb{A}^\omega$ are both used and I would say that they are really essentially different in practice. Both arise in geometric Langlands for example where you want to consider something like the $\C$-vector space $\mathbb C((t))$ as an algebro-geometric object over $\mathbb C$. This has both an ind and a pro direction, which are treated differently when you want to study things like sheaves, D-modules, etc. on such objects. (See work of Raskin, Beraldo...) | |
| Feb 8, 2021 at 20:00 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 8, 2021 at 19:50 | comment | added | Denis Nardin | @TimCampion It might be compact, but it's certainly not proper | |
| Feb 8, 2021 at 19:34 | comment | added | Tim Campion | One thing that confuses me is that I want to object to $Spec k[x_0,x_1,\dots]$ as an infinite-dimensional vector space on the grounds that it is compact, but of course $\mathbb A^1$ is itself compact, which is already confusing. | |
| Feb 8, 2021 at 19:23 | comment | added | Denis Nardin | @TimCampion Well it's also useful to embed 0-dimensional things (or bundles of such), you can get surprisingly much mileage out of that | |
| Feb 8, 2021 at 19:22 | comment | added | Tim Campion | @DenisNardin -- thanks-- I suppose there it's playing more the role of "collecting all the finite-dimensional vector spaces" like topologists do in topological K-theory, rather than being all about embedding things? | |
| Feb 8, 2021 at 19:21 | comment | added | Denis Nardin | @TimCampion In motivic homotopy theory it's the ind-scheme $\mathbb{A}^\infty$ that plays the role usually played by $\mathbb{R}^\infty$. Although, admittedly, trying to embed positive-dimensional varieties in it is problematic. | |
| Feb 8, 2021 at 19:19 | comment | added | Tim Campion | @Wojowu thanks, I intended the "$\cup_n$" to be interpreted as a colimit in the category of schemes, but I am aware that this category lacks many limits and colimits, so you may well be right that it can only be interpreted as an ind-scheme. If both interpretations make sense, then we have potentially 4 infinite-dimensional vector spaces to consider. OTOH I'm not certain -- $Spec k[x_0,x_1,\dots]$ is the product of $\omega$-many copies of $\mathbb A^1$ in the category of affine schemes, but it's not immediate to me that it is so in schemes. I'd worry about ultrafilters somehow creeping in... | |
| Feb 8, 2021 at 19:15 | comment | added | Wojowu | I don't think your $\mathbb A^\infty$ has a natural scheme/variety structure. However, it is an ind-scheme. On the other hand, I'm fairly sure $\operatorname{Spec}k[x_0,\dots]$ is isomorphic to $\mathbb A^\omega$. | |
| Feb 8, 2021 at 18:55 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 8, 2021 at 18:49 | history | asked | Tim Campion | CC BY-SA 4.0 |