# Lifting results from smooth maps to essentially smooth maps.

Recall that a morphism of rings $R\to S$ is called (essentially) smooth if it is formally smooth and (essentially) finitely presented.

(Note: $R\to S$ is essentially finitely presented provided that $S$ is the localization of some finitely
presented $R$-algebra $T$ at some multiplicative system $A \subset T$, that is, $S=A^{-1}T$.)

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?

Edit: According to Mel, every essentially smooth morphism is 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.

• Harry, you should get Mel to register on MO. – Hailong Dao Apr 11 '10 at 4:16
• I feel like you'd have a better chance than I would. =) – Harry Gindi Apr 11 '10 at 4:19
• EGA IV$_4$, 17.5.1(a),(c) – BCnrd Apr 11 '10 at 16:10
• Added a bounty because I couldn't really see how that was at all related to being essentially of finite presentatation. – Harry Gindi Apr 12 '10 at 3:22
• Not saying that it's not, just that I couldn't figure out how to show that they were equivalent, having not read any other parts of EGA 4. – Harry Gindi Apr 12 '10 at 3:23