The Stacks project

Lemma 20.30.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. For every sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ there is a resolution

\[ 0 \to \mathcal{F} \to f_*f^*\mathcal{F} \to f_*f^*f_*f^*\mathcal{F} \to f_*f^*f_*f^*f_*f^*\mathcal{F} \to \ldots \]

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

\[ \mathcal{F}_ x[0] \to \Big( (f_*f^*\mathcal{F})_ x \to (f_*f^*f_*f^*\mathcal{F})_ x \to (f_*f^*f_*f^*f_*f^*\mathcal{F})_ x \to \ldots \Big) \]

is a homotopy equivalence in the category of complexes of $\mathcal{O}_{X, x}$-modules.

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

\[ f^*\mathcal{F}[0] \to \Big( f^*f_*f^*\mathcal{F} \to f^*f_*f^*f_*f^*\mathcal{F} \to f^*f_*f^*f_*f^*f_*f^*\mathcal{F} \to \ldots \Big) \]

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


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