Timeline for Why only finite morphisms in etale fundamental group?
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| Nov 27, 2018 at 11:57 | vote | accept | man | ||
| Nov 26, 2018 at 16:43 | answer | added | M L | timeline score: 9 | |
| Nov 25, 2018 at 20:42 | history | edited | YCor |
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| Nov 25, 2018 at 19:28 | history | edited | man | CC BY-SA 4.0 |
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| Nov 25, 2018 at 19:16 | comment | added | Piotr Achinger | This is sometimes called the "enlarged fundamental group", see SGA3 Exp. X, 6. For a noetherian scheme which is geometrically unibranch (etale locally irreducible), this group agrees with the profinite fundamental group, and for the nodal rational curve it indeed (as expected by Remy) gives $\mathbf{Z}$ rather than $\hat{\mathbf{Z}}$. See also p. 122 in Artin, Mazur "Etale Homotopy". | |
| Nov 25, 2018 at 19:13 | comment | added | R. van Dobben de Bruyn | I think in many contexts there is a natural way to define fundamental groups, but it might not always be exactly the same recipe. For example, Bhatt and Scholze need to look at Noohi groups for their fundamental group. I think with the definition you propose the fundamental group of a projective nodal cubic should be $\mathbb Z$, rather than its usual étale fundamental group $\hat{\mathbb Z}$, as its universal cover is an infinite chain of $\mathbb P^1$s. I believe this has been studied somewhere, but I don't know a reference offhand. Maybe someone will post a genuine answer. | |
| Nov 25, 2018 at 19:01 | comment | added | man | @R.vanDobbendeBruyn I thought of etale coverings that have fibers of countable cardinality. If I understand correctly, such covers would have to be non-quasi-compact. Are you saying that etale fundamental groups can be defined if we allow such covers and the result is the same? I am sorry if the last question is somewhat imprecise, I am only starting to learn this. | |
| Nov 25, 2018 at 18:56 | comment | added | R. van Dobben de Bruyn | What do you mean by infinite étale covers? Étale morphisms are always locally of finite presentation, so the only thing this would allow is certain rising unions (this does lead to a few interesting new covers, but not so many). A more interesting generalisation is to weakly étale morphisms, which leads to the pro-étale topology of Bhatt and Scholze (see also the relevant chapter in the stacks project). But you can never hope to have the exponential $\mathbb C\to\mathbb C^\times$ as a cover, because it is not algebraic. What is it that you're trying to do? | |
| Nov 25, 2018 at 18:43 | history | asked | man | CC BY-SA 4.0 |