I think Ishai's example is close, but one must be a little careful; the normalization of the node is a good example, but the normalization of the cusp is ramified, and the sheaf of relative differentials in that case is not even locally free.

The differential-wise condition you want is this: for the morphism morphism $f: X \to Y$ to be smooth, you need that the sequence

$$0 \to f^* \Omega_Y \to \Omega_X \to \Omega_{X/Y} \to 0$$

be e/ly if $f$ is unramified. But in this case $f^* \Omega_Y \to \Omega_X$ can still fail to be injective.

I think Ishai's example is close, but one must be a little careful; the normalization of the node is a good example, but the normalization of the cusp is ramified, and the sheaf of relative differentials in that case is not even locally free.

The differential-wise condition you want is this: for the morphism morphism $f: X \to Y$ to be smooth, you need that the sequence

$$0 \to f^* \Omega_Y \to \Omega_X \to \Omega_{X/Y} \to 0$$

be exact and locally split (I can't find a reference that says this is sufficient, so it may not be). In the special case when $\dim X = \dim Y$, $\Omega_{X/Y}$ is 0 if and only if $f$ is unramified. But in this case $f^* \Omega_Y \to \Omega_X$ can still fail to be injective.

I think Ishai's example is close, but one must be a little careful; the normalization of the node is a good example, but the normalization of the cusp is ramified, and the sheaf of relative differentials in that case is not even locally free.

The differential-wise condition you want is this: for the morphism morphism f: X --> Y to be smooth, you need that the sequence

0 --> f^* Omega_Y --> Omega_X --> Omega_X/Y --> 0

be exact and locally split (I can't find a reference that says this is sufficient, so it may not be). In the special case when dim X = dim Y, Omega_X/Y is 0 if and only if f is unramified. But in this case f^* Omega_Y --> Omega_X can still fail to be injective.

I think Ishai's example is close, but one must be a little careful; the normalization of the node is a good example, but the normalization of the cusp is ramified, and the sheaf of relative differentials in that case is not even locally free.

The differential-wise condition you want is this: for the morphism morphism $f: X \to Y$ to be smooth, you need that the sequence

$$0 \to f^* \Omega_Y \to \Omega_X \to \Omega_{X/Y} \to 0$$

be e/ly if $f$ is unramified. But in this case $f^* \Omega_Y \to \Omega_X$ can still fail to be injective.