Let $f: X \rightarrow Y$ be a finite type morphism of noetherian schemes with $x \in X$ and $f(x) = y$. Then $f$ is smooth at $x$ if,
1) $f$ is flat at $x$;
2) $\Omega_{X/Y}$ is locally free of rank equal to the krull dimension of $\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{Y, y}} \kappa(y)$.
But when I try to apply this to obtain a definition for a scheme being smooth over a field, it doesn't make sense. If $f:X \rightarrow \text{spec}k$ is the structure morphism, then it is always flat. So we just need to consider the local freeness of $\Omega_{X/k}$. But the residue field of any point in $\text{spec}k$ (indeed the 1 point) is just $k$. So we want $\Omega_{X/k}$ to be locally free with rank at $p \in X$ equal to the krull dimension of $\Omega_{X, p}$. But this is impossible since the rank of a coherent sheaf on a noetherian scheme is locally constant, but the krull dimension of stalks certainly is not.