The Stacks project

Proof. The complex $f_*f^*\mathcal{F} \to f_*f^*f_*f^*\mathcal{F} \to f_*f^*f_*f^*f_*f^*\mathcal{F} \to \ldots$ is the complex associated to the cosimplicial object with terms $f_*f^*\mathcal{F}, f_*f^*f_*f^*\mathcal{F}, f_*f^*f_*f^*f_*f^*\mathcal{F}, \ldots$ described above, see Simplicial, Section 14.25. The augmentation gives rise to the map $\mathcal{F} \to f_*f^*\mathcal{F}$ as indicated. For any abelian sheaf $\mathcal{H}$ on $X_{disc}$ the pushforward $f_*\mathcal{H}$ is flasque because $X_{disc}$ is a discrete space and the pushforward of a flasque sheaf is flasque. Hence the terms of the complex are flasque $\mathcal{O}_ X$-modules.

If $x \in X_{disc} = X$ is a point, then $(f^*\mathcal{G})_ x = \mathcal{G}_ x$ for any $\mathcal{O}_ X$-module $\mathcal{G}$. Hence $f^*$ is an exact functor and a complex of $\mathcal{O}_ X$-modules $\mathcal{G}_1 \to \mathcal{G}_2 \to \mathcal{G}_3$ is exact if and only if $f^*\mathcal{G}_1 \to f^*\mathcal{G}_2 \to f^*\mathcal{G}_3$ is exact (see Modules, Lemma 17.3.1). The result mentioned in the introduction to this section proves the pullback by $f^*$ gives a homotopy equivalence from the constant cosimplicial object $f^*\mathcal{F}$ to the cosimplicial object with terms $f_*f^*\mathcal{F}, f_*f^*f_*f^*\mathcal{F}, f_*f^*f_*f^*f_*f^*\mathcal{F}, \ldots$. By Simplicial, Lemma 14.28.7 we obtain that

is a homotopy equivalence. This immediately implies the two remaining statements of the lemma. $\square$

Proof. Let $\mathcal{A}$ be the category of complexes of $\mathcal{O}_ X$-modules and let $\mathcal{B}$ be the category of complexes of $\mathcal{O}_ X$-modules. Then we can apply the discussion above to the adjoint functors $f^*$ and $f_*$ between $\mathcal{A}$ and $\mathcal{B}$. Arguing exactly as in the proof of Lemma 20.30.1 we get a resolution

in the abelian category $\mathcal{A}$ such that each term of each $f_*f^*\ldots f_*f^*\mathcal{F}^\bullet$ is a flasque $\mathcal{O}_ X$-module and such that for all $x \in X$ the map

is a homotopy equivalence in the category of complexes of complexes of $\mathcal{O}_{X, x}$-modules. Since a complex of complexes is the same thing as a double complex, we can consider the induced map

Since the complex $\mathcal{F}^\bullet$ is bounded below, the same is true for $\mathcal{G}^\bullet$ and in fact each term of $\mathcal{G}^\bullet$ is a finite direct sum of terms of the complexes $f_*f^*\ldots f_*f^*\mathcal{F}^\bullet$ and hence is flasque. The final assertion of the lemma now follows from Homology, Lemma 12.25.5. Since this in particular shows that $\mathcal{F}^\bullet \to \mathcal{G}^\bullet$ is a quasi-isomorphism, the proof is complete. $\square$