Lemma 91.15.1. If $A$ and $B$ are Tor independent $R$-algebras, then the object $E$ in (91.15.0.1) is zero. In this case we have

which is represented by the complex $L_{A/R} \otimes _ R B \oplus L_{B/R} \otimes _ R A $ of $A \otimes _ R B$-modules.

Lemma 91.15.1. If $A$ and $B$ are Tor independent $R$-algebras, then the object $E$ in (91.15.0.1) is zero. In this case we have

\[ L_{A \otimes _ R B/R} = L_{A/R} \otimes _ A^\mathbf {L} (A \otimes _ R B) \oplus L_{B/R} \otimes _ B^\mathbf {L} (A \otimes _ R B) \]

which is represented by the complex $L_{A/R} \otimes _ R B \oplus L_{B/R} \otimes _ R A $ of $A \otimes _ R B$-modules.

**Proof.**
The first two statements are immediate from Lemma 91.6.2. The last statement follows as $L_{A/R}$ is a complex of free $A$-modules, hence $L_{A/R} \otimes _ A^\mathbf {L} (A \otimes _ R B)$ is represented by $L_{A/R} \otimes _ A (A \otimes _ R B) = L_{A/R} \otimes _ R B$
$\square$

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