Timeline for $\Bbb A^1$-Localisation of Schemes, and $\Bbb A^1$-Rigid Schemes
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9 events
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| Jul 21, 2016 at 0:01 | comment | added | Elden Elmanto | Lastly I should also say that canonically mapping your scheme into an A1 invariant (i.e. rigid) scheme is easy: take the albanese. But the hard part is proving that the target is A1-equivalent to the scheme you started with: geometrically doing this is really quite hard! | |
| Jul 20, 2016 at 23:56 | comment | added | Elden Elmanto | Heuristically I think it's pretty impossible to get a simple expression: the localization functor is a left adjoint hence calculated as a colimit. You can ask: "what kind?" as a measure for simplicity. Well you can't do finite coproducts since that clearly doesn't do anything so you are left with sifted ones. You can't just do filtered: spheres are compact so you wont have any interesting homotopy sheaves. You are left with geometric realization: so you need to build a simplicial object resolving your scheme - I cant think of anything easier than Sing! | |
| Jul 20, 2016 at 23:25 | comment | added | Elden Elmanto | There's a general criterion for when you can replace the infinite iteration with just doing it once/twice - see the papers of Asok-Hoyois-Wendt. It works for "(pre)stacky" things as opposed to schemes, but then if you can present your schemes as a fiber of maps between these guys you get the same stoppage of iteration - so, for example, certain homogeneous spaces. Perhaps another class of results of this flavor involves proving that certain schemes are cellular so you can replace just using spheres; the relevant names are Dugger, Isaksen, Wendt and Voelkel. | |
| Jul 20, 2016 at 23:13 | comment | added | user24453 | @elden-elmanto: I am asking if there exists a simplified (not necessary functorial) formula for smooth schemes, as it is the case for $\mathbb{A}^n$ and $\mathbb{A}^1$-rigid schemes. So, is it known what other classes of smooth schemes that admit a relatively simple to express $\mathbb{A}^1$-fibrant replacements, not requiring infinite iterations. I need is compute the hom-sets $[X,Y]_{\mathbb{A}^1}\cong [X,L_{\mathbb{A}^1} Y]$ for some smooth schemes $X,Y$ over a field. I do not know how to make use of the general formulas given in Hirschhorn's book or in Morel-Voevodsky's paper here. | |
| Jul 20, 2016 at 22:01 | comment | added | Elden Elmanto | Oh I guess a countable class of examples not included above are all smooth projective curves of genus $\geq 1$. | |
| Jul 20, 2016 at 21:59 | comment | added | Elden Elmanto | I don't get question 1: the formula $(L_{nis}Sing^{\mathbb{A}^1})^{\omega}$ always works for any simplicial presheaf. For question 2: $\mathbb{A}^1$-rigid schemes have no higher homotopy groups; if $X$ is a monoid scheme then $\Omega BX$ is $\mathbb{A}^1$-local (a "topos"-like property). For a more exotic property: the zero-th $S^1$-slice of the Eilenberg-Maclane $S^1$-spectrum on the free abelian group of an $\mathbb{A}^1$-rigid scheme is not a homotopy invariant presheaf with transfers (see the "An Example" section of Levine's "Slices and Transfers"). | |
| May 26, 2016 at 5:21 | history | edited | user24453 |
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| May 26, 2016 at 4:43 | review | First posts | |||
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| May 26, 2016 at 4:40 | history | asked | user24453 | CC BY-SA 3.0 |