Relative Jacobian condition - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T02:48:35Z http://mathoverflow.net/feeds/question/41048 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41048/relative-jacobian-condition Relative Jacobian condition Makhalan Duff 2010-10-04T18:32:14Z 2010-10-05T01:33:24Z <p>This question is really elementary.</p> <p>Where can I learn about the Jacobian condition not over a field?</p> <p>I think I heard once that there's a Jacobian condition even for $f: X \rightarrow Y$. I'm not really sure what that would be, and what it would imply if it's satisfied, but I'd like some good references so I can learn.</p> http://mathoverflow.net/questions/41048/relative-jacobian-condition/41070#41070 Answer by jlk for Relative Jacobian condition jlk 2010-10-04T21:42:15Z 2010-10-05T01:33:24Z <p>Do you mean that you want a generalization of the Jacobian criteria for the smoothness of, say, a variety to the case of a morphism of schemes? If so, then the <a href="http://books.google.com/books?id=4kgfAQAAIAAJ&amp;q=N%C3%A9ron+models&amp;dq=N%C3%A9ron+models&amp;hl=en&amp;ei=1EmqTMEHxoCUB92TyaUN&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CC0Q6AEwAA" rel="nofollow"> Néron Models </a> book has a nice discussion of this in the section on smoothness.</p>