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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