Timeline for Frobenius base change of etale maps
Current License: CC BY-SA 3.0
11 events
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| Jun 18, 2012 at 15:14 | comment | added | Karl Schwede | Hm, Hugo, I thought about trying to do this using the definition of etale you gave, but I don't see how to do it easily unfortunately. | |
| Jun 16, 2012 at 13:44 | comment | added | Hugo Chapdelaine | Karl, so I've tried to show that the Jacobian's criterion implies flatness and no ramification but it does not seem as easy as what I first thought... | |
| Jun 15, 2012 at 21:29 | comment | added | Karl Schwede | Hugo, you are right, but I don't think I'm doing this, or at least I don't intend to (can you point out to me where?). However, $Fr^{−1}_p(m′)=m$ for obvious reasons (this is the only thing I'm intending to use). In terms your first comment, I've seen Mel Hochster give some variant of the above proof, so that's the one I know. I don't see how to get at it directly from the Jacobian criterion. | |
| Jun 15, 2012 at 21:28 | history | edited | Karl Schwede | CC BY-SA 3.0 |
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| Jun 15, 2012 at 18:50 | comment | added | Hugo Chapdelaine | @Karl, I'm confused about one point, if $A=F_p[x]/x^p$ and $m=(\bar{x})$ then $Fr_p(m)=m'=0$ and therefore is not a maximal ideal... | |
| Jun 15, 2012 at 18:41 | comment | added | Hugo Chapdelaine | I guess that the equivalence of these 2 definitions is probably not too difficult to prove, what do you think? | |
| Jun 15, 2012 at 18:41 | comment | added | Hugo Chapdelaine | Dear @Karl, thanks a lot for your proof, it seems to work. Note though that in your proof you are using another definition of etale namely that the map is flat an unramified. You use the flatness (at least) in order to reduce to the case where $B$ is a free $A$ module (a flat finitely presented modules over a local ring is free) and you use the fact that it is unramified to guarantee that $B/mB$ is a separable field extension of $A/mA$. But in my set up, my definition of etale was different since I used the invertibility of the jacobian. | |
| Jun 15, 2012 at 2:25 | history | edited | Karl Schwede | CC BY-SA 3.0 |
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| Jun 14, 2012 at 22:51 | history | edited | Karl Schwede | CC BY-SA 3.0 |
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| Jun 14, 2012 at 22:23 | history | edited | Karl Schwede | CC BY-SA 3.0 |
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| Jun 14, 2012 at 21:01 | history | answered | Karl Schwede | CC BY-SA 3.0 |