Timeline for The étale fundamental group in the non-normal case
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| May 5, 2013 at 18:39 | comment | added | Andreas Mihatsch | How does this answer xuhan's question? Taking the self-product of two degree n covers from ACL should give a disjoint union of n copies of it. More generally the product of two such covers of degree n and m receives a morphism from the cover of degree lcm(n,m). It is then isomorphic to gcd(n,m) many copies of that. So still every connected component is determined by its degree/generic fibre. | |
| May 5, 2013 at 18:02 | vote | accept | Andreas Mihatsch | ||
| May 5, 2013 at 17:22 | comment | added | Will Sawin | No. Just take the self-product of two copies of ACL's example. There will be many etale covers of each degree. | |
| May 5, 2013 at 15:18 | comment | added | user29283 | It would be instructive to address the related issue of whether a connected finite etale cover of a non-normal irreducible (noetherian) $X$ is determined up to isomorphism by its generic fiber. In ACL's answer, the connected finite etale covers are determined up to isomorphism by their generic fiber. | |
| May 5, 2013 at 13:35 | comment | added | S. Carnahan♦ | As you can see from ACL's example, by losing normality, you lose the correspondence between connected covers and field extensions. The cover gives you an étale algebra over the function field, whose spectrum is disconnected, but the gluing makes the cover itself a connected scheme. | |
| May 5, 2013 at 11:07 | answer | added | ACL | timeline score: 11 | |
| May 5, 2013 at 8:30 | history | asked | Andreas Mihatsch | CC BY-SA 3.0 |