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Let $f:X\rightarrow Y$ a morphism between schemes and $Y'\rightarrow Y$ a fpqc morphism such that the base change $f'$ of $f$ to $Y'$ is formally smooth, does it imply that $f$ is formally smooth?

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The answer to this MO question seems relevant: mathoverflow.net/questions/10731/… –  Alberto García-Raboso Jun 16 '13 at 22:54
    
In the infinitesimal lifting property, can we reduce to easier rings, such local henselian or local for example? –  prochet Jun 17 '13 at 9:42
    
Section 1.7 of arxiv.org/abs/math/9812034 contains the claim that formal smoothness is a local property in the fpqc topology (presumably meaning local on the target), and says that Gabber can explain why. –  S. Carnahan Jul 29 '13 at 6:22
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