The Stacks project

Lemma 21.41.2. Let $\mathcal{C}$ be a site. Let $\mathcal{A}_\bullet \to \mathcal{B}_\bullet $ be a homomorphism of simplicial sheaves of rings on $\mathcal{C}$. If $L\pi _!\mathcal{A}_\bullet \to L\pi _!\mathcal{B}_\bullet $ is an isomorphism in $D(\mathcal{C})$, then we have

\[ L\pi _!(K) = L\pi _!(K \otimes ^\mathbf {L}_{\mathcal{A}_\bullet } \mathcal{B}_\bullet ) \]

for all $K$ in $D(\mathcal{A}_\bullet )$.

Proof. Let $([n], U)$ be an object of $\Delta \times \mathcal{C}$. Since $L\pi _!$ commutes with colimits, it suffices to prove this for bounded above complexes of $\mathcal{O}$-modules (compare with argument of Derived Categories, Proposition 13.29.2 or just stick to bounded above complexes). Every such complex is quasi-isomorphic to a bounded above complex whose terms are flat modules, see Modules on Sites, Lemma 18.28.8. Thus it suffices to prove the lemma for a flat $\mathcal{A}_\bullet $-module $\mathcal{F}$. In this case the derived tensor product is the usual tensor product and is a sheaf also. Hence by Lemma 21.40.2 we can compute the cohomology sheaves of both sides of the equation by the procedure of Lemma 21.40.1. Thus it suffices to prove the result for the restriction of $\mathcal{F}$ to the fibre categories (i.e., to $\Delta \times U$). In this case the result follows from Lemma 21.39.12. $\square$

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