Proposition 90.7.4. Let $A \to B \to C$ be ring maps. There is a canonical distinguished triangle

$L_{B/A} \otimes _ B^\mathbf {L} C \to L_{C/A} \to L_{C/B} \to L_{B/A} \otimes _ B^\mathbf {L} C$

in $D(C)$.

Proof. Consider the short exact sequence of sheaves of Lemma 90.7.1 and apply the derived functor $L\pi _!$ to obtain a distinguished triangle

$L\pi _!(g_1^{-1}\Omega _1 \otimes _{\underline{B}} \underline{C}) \to L\pi _!(g_2^{-1}\Omega _2) \to L\pi _!(g_3^{-1}\Omega _3) \to L\pi _!(g_1^{-1}\Omega _1 \otimes _{\underline{B}} \underline{C})$

in $D(C)$. Using Lemmas 90.7.3 and 90.4.3 we see that the second and third terms agree with $L_{C/A}$ and $L_{C/B}$ and the first one equals

$L\pi _{1, !}(\Omega _1 \otimes _{\underline{B}} \underline{C}) = L\pi _{1, !}(\Omega _1) \otimes _ B^\mathbf {L} C = L_{B/A} \otimes _ B^\mathbf {L} C$

The first equality by Cohomology on Sites, Lemma 21.38.6 (and flatness of $\Omega _1$ as a sheaf of modules over $\underline{B}$) and the second by Lemma 90.4.3. $\square$

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