Corollary 3

Let $k$ be a field of characteristic 0 and $A$ a $k$-algebra essentially of finite type. For $A$ to be regular, it is necessary and sufficient that the $A$-module $\Omega_k(A)$ be projective.

By Corollary 2, it is enough to prove that an artinian local $k$-algebra $A$ such that $\Omega_k(A)$ is a free $A$-module is a field. Now, by IX, § 3, No. 3, Theorem 1, there exists a subfield $K$ of $A$ such that $A = K \oplus m_A$. Let $\delta : A \to m_A/m_A^2$ be the composite mapping of the projection of $A$ onto $m_A$ and the canonical mapping $m_A \to m_A/m_A^2$. One verifies as in the proof of Lemma 4 that $\delta$ is a $k$-derivation. But every $k$-derivation of $A$ into an $A$-module annihilates $m_A$: let in fact $x \in m_A$, and $n$ an integer $\geq 1$ such that one has $x^{n-1} \neq 0,\ x^n = 0$. This implies in the $A$-module $\Omega_k(A)$ the relation $nx^{n-1} dx = 0$, hence $dx = 0$ since $nx^{n-1} \neq 0$. Thus $\delta(m_A) = 0$, hence $m_A = m_A^2$, and finally $m_A = \{0\}$.