Corollary 3

Let $k$ be a ring and $A$ a formally smooth $k$-algebra.

a) The $A$-module $\Omega_k(A)$ is projective.

b) Suppose that the ring $A \otimes_k A$ is noetherian. Let $\mu : A \otimes_k A \to A$ denote the homomorphism such that $\mu(x \otimes y) = xy$; then the ideal $\mathrm{Ker}(\mu)$ is completely secant.

The $k$-algebras $A$ and $A \otimes_k A$ are formally smooth (No. 2, prop. 4, c)), and $A$ is isomorphic to the quotient of $A \otimes_k A$ by the kernel $I$ of $\mu$. By definition one has $\Omega_k(A) = I/I^2$. Thus a) and b) result from th. 2.