Timeline for Comparing etale covers of a curve on two different algebraically closed fields $K \subseteq L$
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
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| Oct 23, 2010 at 0:57 | comment | added | James D. Taylor | In case you know, where can I find a proof (preferably in English, but French is fine) of this map between fundamental groups being an isomorphism? I've seen similar proofs, but not quite that. | |
| Oct 23, 2010 at 0:06 | comment | added | BCnrd | Ramification at the missing points is tame, and the discrepancy between the $\pi_1$'s of the given curves and their compactifications is governed by the (tame!) inertia groups at the missing points. Since tame inertia doesn't notice $L/K$, one can make covers sufficiently ramified at missing points and etale over $C$ that "eat up" ramification for the covering of $C_L$, and so by the Galois correspondence between group theory and intermediate curves reduce to everywhere unramified covers of certain curves over $C$. Then SGA1 results on specialization of $\pi_1$ in proper case can be applied. | |
| Oct 22, 2010 at 23:21 | comment | added | Pete L. Clark | Yes, it's true, and at least morally I think it's equivalent to the statement about the natural map on $\pi_1$'s being an isomorphism. Someone else will probably come along with a more detailed answer, but in the meantime have you tried looking in Szamuely's book Galois groups and fundamental groups? | |
| Oct 22, 2010 at 20:53 | history | asked | James D. Taylor | CC BY-SA 2.5 |