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The Stacks project

Lemma 10.131.7. Let A \to B \to C be ring maps. Then there is a canonical exact sequence

C \otimes _ B \Omega _{B/A} \to \Omega _{C/A} \to \Omega _{C/B} \to 0

of C-modules.

Proof. We get a diagram (10.131.4.1) by putting R = A, S = C, R' = B, and S' = C. By Lemma 10.131.6 the map \Omega _{C/A} \to \Omega _{C/B} is surjective, and the kernel is generated by the elements \text{d}(c), where c \in C is in the image of B \to C. The lemma follows. \square


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