A morphism of schemes is formally smooth and locally of finite presentation iff it is smooth. 


What happens if we drop the finitely presented hypothesis? Of course, locally of finite presentation is part of smoothness, so implicilty I am asking for the flatness to fail.



Is there an example of a formally smooth morphism which is not smooth?
Here's an elementary example. For any field $k$, consider the ring $k[t^q|q\in\mathbb Q\_{>0}]$, which I'll abbreviate $k[t^q]$. I claim that the natural quotient $k[t^q]\to k$ given by sending $t^q$ to $0$ is formally smooth but *not flat*, and therefore not smooth.


First let's show it's formally smooth. Let $A$ be a ring with square-zero ideal $I\subseteq A$, and suppose we have maps $f:k[t^q]\to A$ and $g:k\to A/I$ making the following square commute (I drew it backwards because you're probably thinking of Spec of everything)


$$
\begin{array}{ccc}
 A/I & \xleftarrow g & k \\
 \uparrow & & \uparrow\\
 A & \xleftarrow f & k[t^q]
\end{array}
$$


We'd like to show that there's a map $k\to A$ filling the diagram in. For any $q\in \mathbb Q\_{>0}$, note that $f(t^q)\in I$ by commutativity of the square, so $f(t^{2q})\in I^2=0$. But every $q$ is of the form $2q'$ for some $q'$, so we've shown that $f(t^q)=0$ for all $q\in \mathbb Q\_{>0}$. So $f$ factors through $k$, as desired.


Now let's show that $k$ is not flat over $k[t^q]$. Consider the exact sequence 
$$0\to (t)\to k[t^q]\to k[t^q]/(t)\to 0.$$
When you tensor with $k$, you get 
$$0\to k\to k\to k\to 0,$$
which is obviously not exact. So $k$ is not flat over $k[t^q]$.


