Is this result well-known (or even true)? Let \begin{align*} f:A & \to B\\ g:A & \to C \end{align*} be homomorphisms of finitely generate $k$-algebras and let

$R=B\otimes_{k}C$ and $S=B\otimes_{A}C$

Then there is a surjective homomorphism $$ \xi:R\to S $$ and $$ \Omega_{S/k}=S\otimes_{B}\Omega_{B/k}\oplus_{\Omega_{A/k}}S\otimes_{C}\Omega_{C/k} $$ Where the $\oplus$ with a subscript represents a push-out.

Algebraic geometry, R. Hartshorne. Second, $\Omega_{S/B}$ is isomorphic to $S\otimes_C \Omega_{C/A}$, resp. $\Omega_{S/A}$ is isomorphic to $S\otimes_B \Omega_{B/A}$, cf. Proposition 8.2A. Finally, using Proposition 8.3A once more, the result follows. $\endgroup$ – Jason Starr Mar 26 '18 at 17:29