Lemma 92.6.2. If (92.6.0.1) induces a quasi-isomorphism $B \otimes _ A^\mathbf {L} A' = B'$, then the functoriality map induces an isomorphism
Proof. We will use the notation introduced in Equation (92.6.0.2). We have
the first equality by Lemma 92.4.3 and the second by Cohomology on Sites, Lemma 21.39.6. Since $\Omega _{\mathcal{O}/A}$ is a flat $\mathcal{O}$-module, we see that $\Omega _{\mathcal{O}/A} \otimes _\mathcal {O} \underline{B}$ is a flat $\underline{B}$-module. Thus $Lh^*(\Omega _{\mathcal{O}/A} \otimes _\mathcal {O} \underline{B}) = \Omega _{\mathcal{O}/A} \otimes _\mathcal {O} \underline{B'}$ which is equal to $g^{-1}(\Omega _{\mathcal{O}'/A'} \otimes _{\mathcal{O}'} \underline{B'})$ by inspection. we conclude by Lemma 92.6.1 and the fact that $L_{B'/A'}$ is computed by $L\pi '_!(\Omega _{\mathcal{O}'/A'} \otimes _{\mathcal{O}'} \underline{B'})$. $\square$
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