Lemma 10.150.7. Let $R \to S \to S'$ be ring maps. Let $J$, resp. $J'$ be the kernel of the multiplication map $S \otimes _ R S \to S$, resp. $S' \otimes _{R'} S' \to S'$. If $S \to S'$ is formally étale, then the map
is an isomorphism for all $k \geq 0$. In particular, the map $S' \otimes _ S \Omega _{S/R} \to \Omega _{S'/R}$ is an isomorphism.
Proof. Observe that $S' \otimes _ S (S \otimes _ R S) = S' \otimes _ R S$ and the ideal $J$ generates in $S' \otimes _ R S$ the kernel $I$ of the surjection $S' \otimes _ R S \to S'$. Whence the left hand side is equal to $S' \otimes _ R S/I^{k + 1}$. The map $S' \otimes _ R S \to S' \otimes _ R S'$ is formally étale by Lemma 10.150.2. Thus we conclude that the displayed arrow in the statement of the lemma is an isomorphism by Lemma 10.150.6. The final assertion follows from this and the fact that $\Omega _{S/R} = J/J^2$ and $\Omega _{S'/R} = J'/(J')^2$ by Lemma 10.131.13. $\square$
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