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.

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