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The Stacks project

Lemma 10.134.7. Let A \to B \to C be ring maps. Assume A \to C is surjective (so also B \to C is). Denote I = \mathop{\mathrm{Ker}}(A \to C) and J = \mathop{\mathrm{Ker}}(B \to C). Then the sequence

I/I^2 \to J/J^2 \to \Omega _{B/A} \otimes _ B B/J \to 0

is exact.

Proof. Follows from Lemma 10.134.4 and the description of the naive cotangent complexes \mathop{N\! L}\nolimits _{C/B} and \mathop{N\! L}\nolimits _{C/A} in Lemma 10.134.6. \square


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