Timeline for Is there an example of a formally smooth morphism which is not smooth?
Current License: CC BY-SA 3.0
6 events
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| Sep 5, 2011 at 23:13 | comment | added | Akhil Mathew | I'm just going to point out a small extension of this, and simultaneously advertise a little. In general, if $R$ is any ring and $I \subset R$ an ideal with $I^2 =I$, then $R \to R/I$ is formally \'etale (but probably not flat, i.e. unless $I$ is a direct factor). This follows by the same argument, and is Prop. 2.20 in people.fas.harvard.edu/~amathew/chetale.pdf Note that if $R$ is noetherian, then in this situation $R /I$ is always flat over $R$ (as a formally smooth algebra over a noeth. ring is flat) because $I$ is automatically a direct factor. | |
| Sep 5, 2011 at 22:59 | history | edited | Anton Geraschenko | CC BY-SA 3.0 |
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| Jan 17, 2010 at 22:39 | comment | added | Georges Elencwajg | Great, crisp answer, Anton! And as a bonus it shows, as Mabli has just remarked, that even formally étale doesn't imply flat (since your example is formally étale). | |
| Oct 8, 2009 at 15:14 | vote | accept | David Zureick-Brown | ||
| Oct 8, 2009 at 15:13 | vote | accept | David Zureick-Brown | ||
| Oct 8, 2009 at 15:14 | |||||
| Oct 8, 2009 at 14:50 | history | answered | Anton Geraschenko | CC BY-SA 2.5 |