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A variation on Ishaii's 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)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_B = Omega(A) Omega_A (x) B / d(e^2) = k[e] / (e^2, 2e)


via the isomorphism Omega(A) 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} / I(B/A)^2 I_{B/A}^2 ---> Omega(A) --> 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) I_{B/A} for the ideal of B in A.A.) Even in characteristic 2, the differential has a kernel, so the cotangent complex is not concentrated in degree 0.

1

A variation on Ishaii'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.