Lifting results from smooth maps to essentially smooth maps. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T01:24:54Z http://mathoverflow.net/feeds/question/20890 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20890/lifting-results-from-smooth-maps-to-essentially-smooth-maps Lifting results from smooth maps to essentially smooth maps. Harry Gindi 2010-04-10T03:13:33Z 2010-04-15T09:14:22Z <p>Recall that a morphism of rings $R\to S$ is called (essentially) <em>smooth</em> if it is formally smooth and (essentially) finitely presented.</p> <p>(Note: $R\to S$ is <em>essentially finitely presented</em> provided that $S$ is the localization of some finitely<br> presented $R$-algebra $T$ at some multiplicative system $A \subset T$, that is, $S=A^{-1}T$.)</p> <p>In class, our professor said that working with smooth or essentially smooth morphisms yields an effectively equivalent theory. This motivates my question: Is there a general technique to lift results from the smooth case to the essentially smooth case?</p> <p>Edit: According to Mel, every essentially smooth morphism <em>is</em> a localization of a smooth morphism. However, this direction is much more involved than the other direction, which is immediate from the definitions. Anyway, this would be the answer to the question. </p> http://mathoverflow.net/questions/20890/lifting-results-from-smooth-maps-to-essentially-smooth-maps/21436#21436 Answer by Gjergji Zaimi for Lifting results from smooth maps to essentially smooth maps. Gjergji Zaimi 2010-04-15T09:14:22Z 2010-04-15T09:14:22Z <p>It seems that what you are looking for is theorem 5.11 <a href="http://arxiv.org/abs/0809.1201" rel="nofollow">here</a>. See also example (e) on section 5.12. Also if you don't feel like reviewing from EGA you can look at section 1.5 of "Introduction to algebraic stacks" by A. Canonaco which I think covers the relevant facts (including 17.5.1)</p>