Lemma 90.8.6. Let $A \to B$ be ring maps and $S \subset A$, $T \subset B$ multiplicative subsets such that $S$ maps into $T$. Then $L_{T^{-1}B/S^{-1}A} = L_{B/A} \otimes _ B T^{-1}B$ in $D(T^{-1}B)$.

**Proof.**
Lemma 90.8.5 shows that $L_{T^{-1}B/A} = L_{B/A} \otimes _ B T^{-1}B$ and Lemma 90.8.1 shows that $L_{T^{-1}B/A} = L_{T^{-1}B/S^{-1}A}$.
$\square$

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