Lemma 10.134.11. Let $S \subset A$ is a multiplicative subset of $A$. Let $S^{-1}A \to B$ be a ring map. Then $\mathop{N\! L}\nolimits _{B/A} \to \mathop{N\! L}\nolimits _{B/S^{-1}A}$ is a homotopy equivalence.

**Proof.**
Choose a presentation $\alpha : P \to B$ of $B$ over $A$. Then $\beta : S^{-1}P \to B$ is a presentation of $B$ over $S^{-1}A$. A direct computation shows that we have $\mathop{N\! L}\nolimits (\alpha ) = \mathop{N\! L}\nolimits (\beta )$ which proves the lemma as the naive cotangent complex is well defined up to homotopy by Lemma 10.134.2.
$\square$

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