It seems to be well-known that for a morphism of schemes that flat and unramified is equivalent to smooth and unramified, and we call this notion etale. I am having a little trouble proving this result, particularly when it comes to the fact that these notions are only defined to be locally of finite presentation.
Let $f:X \rightarrow Y$ be a morphism of schemes. If it is smooth and unramified then it is flat and unramified trivially.
Conversely, if it is flat and unramified, we want to show smoothness. Taking our definition as being locally of finite presentation, flat, and geometrically regular, it remains only to show that it is geometrically regular.
To this end, consider a base change $\operatorname{spec} \overline{\kappa(f(p))} \rightarrow Y$ where $p$ is a point of $X$ and $\overline{\kappa(f(p))}$ is the algebraic closure of its residue field. Then consider the fibered product scheme $\overline{X_{f(p)}}$. Since $f$ is unramified, we can pullback the sheaf of differentials to find that
$$\Omega_{\overline{X_{f(p)}}/\overline{\kappa(f(p))}} = 0.$$ Now we use the following fact: If $A$ is a local ring containing a field $k$ which is isomorphic to its residue field, and if furthermore $B$ is a localisation of a finitely generated $k$-algebra, then B is regular if and only if the module of differentials $\Omega_{B/k}$ is free.
Now if we had set $f$ to be of finite presentation then we would be done. Indeed the pullback would be of finite presentation, and so $\overline{X_{f(p)}}$ would be quasicompact. Then every closed point lies in the closure of a non-closed point. Then we can apply the above result from commutative algebra, along with the fact that any localisation of a regular local ring remains regular.
However since $\overline{X_{f(p)}}$ is not necessarily quasi-compact, it seems we can only deduce regularity at closed points, which in this case is not enough to deduce it about every point in $\overline{X_{f(p)}}$. Is there a way around this, or does the equivalence of smooth & unramified with etale & unramified only hold in the quasicompact case?