Timeline for What is the definition of smoothness in scheme theory?
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14 events
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| May 7, 2021 at 4:43 | comment | added | user196717 | If we do $\mathbb{A}^1$-localization then of course we have $H^1_{\mathrm{mot}}(\mathbb{A}^1,\mathscr{F})=0$ but I'm uncomfortable with that fact "affine spaces may not be contractible with étale cohomology sense"... | |
| May 7, 2021 at 4:35 | comment | added | user196717 | To eliminate Artin-Schreier covering, we can do perfectoidification of $\mathbb{A}^1$! and my proof of characterization of smooth scheme is essentially same as stacks project proof you gave. But even after perfectoidification, I think $\mathbb{A}^1_{\mathrm{perfd}}$ is not so same as $\mathbb{R}^1$. This is my main problem. | |
| May 7, 2021 at 4:31 | comment | added | user196717 | Of course, any smooth scheme is étale locally looks like affine space. But the problem, in my opinion, is that affine spaces are not so same as Euclidean space. For example, let $k$ be a char. $p$ algebraically closed field and $X=\mathbb{A}^1$ be an affine line. Then for $p\ne \ell$, using Artin-Schreier covering, we have that there is a locally constant sheaf $\mathscr{F}$ over $X$ with invertible $p$ such that $H^1(\mathbb{A}^1,\mathscr{F})\ne 0$! (In char. $0$ case, using Hurwitz formula, we have $H^1(\mathbb{A}^1,\mathscr{F})=0$ for all locally constant $\mathscr{F}$.) | |
| May 6, 2021 at 19:28 | comment | added | François Brunault | I don't know if this is what you are after, but smooth morphisms of schemes are characterised as those which étale-locally look like affine spaces. See stacks.math.columbia.edu/tag/054L In my view this is the right analogue of the manifold definition. (Of course this doesn't hold with the Zariski topology.) | |
| May 5, 2021 at 9:45 | comment | added | Denis Nardin | @Apjoo I think you are referring to the fact that a Noetherian ring is regular iff every pseudocoherent complex is perfect (this is a globalized version of the theorem of Serre that a Noetherian local ring is regular iff it has finite global dimension). But Mizi seems to be after a characterization in terms of constructible sheaves, not coherent.. | |
| May 5, 2021 at 9:14 | comment | added | user196717 | Then it seems the notion of smoothness is much close to locally Euclidean property. But Euclidean spaces have much good properties than perfectoid space such as vanishing of higher homotopy... (I don't know how to compute algebraic K-theory even when X is just a perfectoid point.) | |
| May 5, 2021 at 8:57 | comment | added | Apjoo | Actually there is a theorem of Serre describing regular rings among Noetherian local rings in terms of modules. I don't know if that's what you want. | |
| May 5, 2021 at 8:54 | comment | added | Denis Nardin | @Mizi Every CW complex is locally contractible. In particular the complex (or real) points of an algebraic variety are always locally contractible, even when the variety is not smooth. For example the wedge of two circles is locally contractible, but it's quite far from being locally Euclidean. | |
| May 5, 2021 at 8:48 | comment | added | user196717 | Then the class of locally contractible spaces is larger than the class of locally Euclidean space. Sorry for my poor general topology knowledge. | |
| May 5, 2021 at 8:42 | comment | added | user196717 | Thanks! then my quasi-conjecture may fails. (But we can ask that if we are considering any constructible sheaves not just $\mathbb{Q}_{\ell}(i)$ then can the "linear" notion controls smoothness...?) | |
| May 5, 2021 at 8:39 | history | edited | user196717 | CC BY-SA 4.0 |
added 34 characters in body
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| May 5, 2021 at 8:17 | comment | added | Apjoo | It seems unlikely that smoothness is equivalent a "linear" notion (defined in terms of derived categories, cohomology) see mathoverflow.net/q/18006/201955 | |
| May 5, 2021 at 8:16 | comment | added | Apjoo | Not every locally contractible space is locally Euclidean see arxiv.org/abs/1201.3897 | |
| May 5, 2021 at 7:22 | history | asked | user196717 | CC BY-SA 4.0 |