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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?

For instance, if $R$ is reduced

Edit: According to Mel, and $R\to S$ every essentially smooth morphism is a localization of a smooth morphism. However, then $S$ this direction is reduced. It turns out that much more involved than the same fact other direction, which is true for essentially smooth mapsimmediate from the definitions. Is there a good way to "lift" Anyway, this result would be the answer to the more general case?question.

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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 way technique to lift results from the smooth case to the essentially smooth case?

For instance, if $R$ is reduced, and $R\to S$ is smooth, then $S$ is reduced. It turns out that the same fact is true for essentially smooth maps. Is there a good way to "lift" this result to the more general case?

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# DoessentiallyLiftingresultsfrom smooth ring maps factorintoacompositionofatoessentially smooth mapandalocalization?maps.

A

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

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

Can we factor an S=A^{-1}T$.) In class, our professor said that working with smooth or essentially smooth map into morphisms yields an effectively equivalent theory. This motivates my question: Is there a general way to lift results from the smooth map and a localization? Certainly case to the converse essentially smooth case? For instance, if$R$is truereduced, i.e. the localization of any smooth and$R$-algebra R\to S$ is essentially smooth, since localizations are formally étale (except of course when then $S$ is reduced. It turns out that the multiplicative system contains zero.)

If not, are there any conditions same fact is true for when we canessentially smooth maps. Is there a good way to "lift" this result to the more general case?