Formally étale means that the infinitesimal lifting property is uniquely satisfied. If the map is also locally of finite presentation, then it is called étale. One of many characterizations (see EGA 4.5.17) of étale is flat and unramified. So my question is whether the weaker condition of formally étale still implies flatness?
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$\begingroup$ I don't know if your question is answered there, but you might want to take a look at the answers to mathoverflow.net/questions/8451/definition-of-etale-for-rings $\endgroup$– Charles RezkCommented Jan 15, 2010 at 18:58
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2 Answers
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It seems that Anton Geraschenkos answer to a previous question Is there an example of a formally smooth morphism that is not smooth does the trick here as well. His example of a formally smooth map that is not flat is indeed formally etale. So, formally etale does not imply flat.
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A formally smooth ring map between Noetherian rings is flat, see https://stacks.math.columbia.edu/tag/07NP.
