Timeline for Discretifications of the fundamental group functor
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11 events
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| Aug 9, 2016 at 22:03 | history | edited | Myshkin | CC BY-SA 3.0 |
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| Aug 22, 2012 at 13:55 | comment | added | o a | Will Savin: yes, it is exactly the point of the question to "go around" that counterexample. I should perhaps add that the reason for the question is that one can prove very partial positive results, for some very small subcategories of Schemes. | |
| Aug 22, 2012 at 13:53 | history | edited | o a | CC BY-SA 3.0 |
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| Aug 21, 2012 at 20:43 | comment | added | Will Sawin | I see. I thought it wouldn't work for $K\subset \mathbb C$ because of the classic example that shows the fundamental group of a complex manifold is not determined by its algebraic structure, but that is just an example of the $Aut(\mathbb C/\mathbb Q)$ action on discretization. In characteristic $p$, The Artin-Schreier sequence shows that $H^1(X,\mathbb Z/p)= Mor(\pi_1(X),\mathbb Z/p)$ is infinite-dimensional for a general scheme, so $\pi_1(x)$ is infinitely generated, which doesn't rule out a by-hand construction but makes it seem difficult. | |
| Aug 21, 2012 at 19:04 | comment | added | o a | Will Savin: the topological fundamental group is intended to be an example. it works for K=C or K⊂C and an appropriate subcategory of Schemes. (Well, the topological fundamental group of a smooth projective complex algebraic variety is not necessarily residually finite, so you have to do something about that). In general, I do not know; perhaps my definition is too naive to work for general Schemes. Also, in char p, I do not know; but possibly it is not too hard to construct by hand. | |
| Aug 21, 2012 at 17:32 | comment | added | Will Sawin | Is it obvious whether there exists any discretification? | |
| Aug 21, 2012 at 17:31 | history | edited | o a | CC BY-SA 3.0 |
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| Aug 21, 2012 at 13:00 | history | edited | o a |
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| Aug 21, 2012 at 12:33 | comment | added | o a | Thanks. Yes, this kind of question is probably indirectly related...But I do not see a direct relation. | |
| Aug 21, 2012 at 11:41 | comment | added | HJRW | Possibly related question (or not - I'm out of my depth): mathoverflow.net/questions/4972/… . | |
| Aug 21, 2012 at 9:22 | history | asked | o a | CC BY-SA 3.0 |