Lemma 21.39.12. Let $\mathcal{C}$ be a category (endowed with chaotic topology). Let $\mathcal{O} \to \mathcal{O}'$ be a map of sheaves of rings on $\mathcal{C}$. Assume

1. there exists a cosimplicial object $U_\bullet$ in $\mathcal{C}$ as in Lemma 21.39.7, and

2. $L\pi _!\mathcal{O} \to L\pi _!\mathcal{O}'$ is an isomorphism.

For $K$ in $D(\mathcal{O})$ we have

$L\pi _!(K) = L\pi _!(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}')$

in $D(\textit{Ab})$.

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

$S_\bullet = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_\bullet , U)$

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

$X_\bullet : n \longmapsto \bigoplus \nolimits _{s \in S_ n} P_ n$

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

$X'_\bullet : n \longmapsto \bigoplus \nolimits _{s \in S_ n} P'_ n$

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. $\square$

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