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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 $Y$?

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You write "as every thing is locally of finite type", but prior to that you make no such lft hypothesis, so it is confusing. If $X$ and $Y$ are locally finite type over $k$ then any $k$-morphism between them is lft (equivalently, locally of finite presentation), and for an lfp map smoothness is the same as formal smoothness by EGA. So in the lft case, if f is formally smooth then it is smooth, and thus flat. So then $Y$ inherits $k$-smoothness from $X$, as we can pass up to $\overline{k}$ and use that a noetherian ring is regular if it has a faithfully flat extension ring that is regular. –  user29720 Jun 23 '13 at 16:51
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what I meant is that in the lft case it's well known but I would like to know if we can weaken these hypotheses. –  prochet Jun 23 '13 at 16:53
    
By Theorem 28.7 in Matsumura's "Commutative Ring Theory", a noetherian local $k$-algebra is formally $k$-smooth for its max-adic topology if and only if it is "geometrically regular" over $k$, in the sense that it remains regular after tensoring against any finite extension of $k$. By 19.7.1 in EGA 0$_{\rm{IV}}$, if $X$ and $Y$ are locally noetherian and $f$ between local rings is formally smooth for max-adic topologies then it is 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
    
so it's probably true if X and Y are locally noetherian? To relate global to local, maybe it's sufficient to say that formal smoothness is local for Zariski topology on the source and on the target by 17.1.6 (i) and (ii) in EGA IV.4 –  prochet Jun 23 '13 at 22:32
    
and let say that only Y is locally noetherian and moreover the morphism is faithfully flat? –  prochet Jun 23 '13 at 22:34
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