The Stacks project

Proof. Note: in this proof $L\pi _!$ denotes the left derived functor of $\pi _!$ on abelian sheaves. 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 direct sums of $j_{U!}\mathcal{O}_ U$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, see Modules on Sites, Lemma 18.28.8. Thus it suffices to prove the lemma for $j_{U!}\mathcal{O}_ U$. By assumption

is a simplicial set homotopy equivalent to the constant simplicial set on a singleton. Set $P_ n = \mathcal{O}(U_ n)$ and $P'_ n = \mathcal{O}'(U_ n)$. Observe that the complex associated to the simplicial abelian group

computes $L\pi _!(j_{U!}\mathcal{O}_ U)$ by Lemma 21.39.7. Since $j_{U!}\mathcal{O}_ U$ is a flat $\mathcal{O}$-module we have $j_{U!}\mathcal{O}_ U \otimes ^\mathbf {L}_\mathcal {O} \mathcal{O}' = j_{U!}\mathcal{O}'_ U$ and $L\pi _!$ of this is computed by the complex associated to the simplicial abelian group

As the rule which to a simplicial set $T_\bullet $ associated the simplicial abelian group with terms $\bigoplus _{t \in T_ n} P_ n$ is a functor, we see that $X_\bullet \to P_\bullet $ is a homotopy equivalence of simplicial abelian groups. Similarly, the rule which to a simplicial set $T_\bullet $ associates the simplicial abelian group with terms $\bigoplus _{t \in T_ n} P'_ n$ is a functor. Hence $X'_\bullet \to P'_\bullet $ is a homotopy equivalence of simplicial abelian groups. By assumption $P_\bullet \to P'_\bullet $ is a quasi-isomorphism (since $P_\bullet $, resp. $P'_\bullet $ computes $L\pi _!\mathcal{O}$, resp. $L\pi _!\mathcal{O}'$ by Lemma 21.39.7). We conclude that $X_\bullet $ and $X'_\bullet $ are quasi-isomorphic as desired. $\square$