Lemma 86.3.3. With assumptions as in Lemma 86.3.2 assume that $B/I^ nB \to C/I^ nC$ is a local complete intersection homomorphism for all $n$. Then $H^{-1}(\mathop{N\! L}\nolimits ^\wedge _{B/A} \otimes _ B C) \to H^{-1}(\mathop{N\! L}\nolimits ^\wedge _{C/A})$ is injective.

**Proof.**
By More on Algebra, Lemma 15.33.6 we see that this holds for the map between naive cotangent complexes of the situation modulo $I^ n$ for all $n$. In other words, we obtain a distinguished triangle in $D(C/I^ nC)$ for every $n$. Using Lemma 86.3.1 this implies the lemma; details omitted.
$\square$

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