A variation on Ishai's example is a closed embedding: its sheaf of relative differentials is `0`

, hence free of finite rank, even though it needn't be smooth.

However, `k[e] / e^2`

over `k`

is not actually a counterexample (except in characteristic `2`

). The module of relative differentials of `Spec k[e] / e^2`

over `Spec k`

is not free if the characteristic of `k`

is not 2. Let `A = k[e]`

and `B = k[e] / e^2`

. Then

```
Omega_B = Omega_A (x) B / d(e^2) = k[e] / (e^2, 2e)
```

via the isomorphism `Omega_A --> A : dt --> 1`

. This is not isomorphic to `B`

unless `2 = 0`

.

On the other hand, you can conclude that `B`

is smooth if its cotangent complex is a vector bundle in degree 0. In the case of `k[e] / e^2`

, the cotangent complex is

```
[ I_{B/A} / I_{B/A}^2 ---> Omega_A (x) B ] = [ e^2 A / e^4 A ---> B de ]
```

in degrees `[-1,0]`

and the differential is the universal derivation. (I write `I_{B/A}`

for the ideal of `B`

in `A`

.) Even in characteristic `2`

, the differential has a kernel, so the cotangent complex is not concentrated in degree `0`

.