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 $L_{A/R} \otimes _ A (A \otimes _ R B) = L_{A/R} \otimes _ R B$ $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).