Lemma 10.134.10. If $S \subset A$ is a multiplicative subset of $A$, then $\mathop{N\! L}\nolimits _{S^{-1}A/A}$ is homotopy equivalent to the zero complex.

Proof. Since $A \to S^{-1}A$ is flat we see that $\mathop{N\! L}\nolimits _{S^{-1}A/A} \otimes _ A S^{-1}A \to \mathop{N\! L}\nolimits _{S^{-1}A/S^{-1}A}$ is a homotopy equivalence by flat base change (Lemma 10.134.8). Since the source of the arrow is isomorphic to $\mathop{N\! L}\nolimits _{S^{-1}A/A}$ and the target of the arrow is zero (by Lemma 10.134.6) we win. $\square$

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