Lifting results from smooth maps to essentially smooth maps.

2010-04-10T03:13:33Z

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.

Answer by Gjergji Zaimi for Lifting results from smooth maps to essentially smooth maps.

2010-04-15T09:14:22Z

It seems that what you are looking for is theorem 5.11 here. 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)

Lifting results from smooth maps to essentially smooth maps. - MathOverflow

Lifting results from smooth maps to essentially smooth maps.

Answer by Gjergji Zaimi for Lifting results from smooth maps to essentially smooth maps.