Let $f:X\rightarrow Y$ a morpism between $k$-schemes ( $k$ a field).

We suppose that X is formally smooth and f is formally smooth and surjective.

Do we have that $Y$ is formally smooth?

Or if it's not is there weaker hypotheses as every thing is locally of finite type to obtain smoothness of $Y$?

flat, so $Y$ inherits geometric regularity over $k$ from $X$. So try to relate "global" formal smoothness to max-adic formal smoothness on local rings. – user29720 Jun 23 '13 at 19:19