Hartshorne, Algebraic Geometry
In example III.10.0.3, Hartshorne remarks that with k algebraically closed, X smooth of dimension n over Spec k is equivalent to X regular of dimension n. He references II.8.8.
However II.8.8 requires that when one looks at a local ring B, the relative sheaf of differentials for this local ring must be a free B-module. This is certainly true if X is irreducible. But what if X is not irreducible?
So my overall question: is the statement from III.10.0.3 true for X not irreducible?