## 20.30 Godement resolution

A reference is [Godement].

Let $(X, \mathcal{O}_ X)$ be a ringed space. Denote $X_{disc}$ the discrete topological space with the same points as $X$. Denote $f : X_{disc} \to X$ the obvious continuous map. Set $\mathcal{O}_{X_{disc}} = f^{-1}\mathcal{O}_ X$. Then $f : (X_{disc}, \mathcal{O}_{X_{disc}}) \to (X, \mathcal{O}_ X)$ is a flat morphism of ringed spaces. We can apply the dual of the material in Simplicial, Section 14.34 to the adjoint pair of functors $f^*, f_*$ on sheaves of modules. Thus we obtain an augmented cosimplicial object

$\xymatrix{ \text{id} \ar[r] & f_*f^* \ar@<1ex>[r] \ar@<-1ex>[r] & f_*f^*f_*f^* \ar@<0ex>[l] \ar@<-2ex>[r] \ar@<0ex>[r] \ar@<2ex>[r] & f_*f^*f_*f^*f_*f^* \ar@<1ex>[l] \ar@<-1ex>[l] }$

in the category of functors from $\textit{Mod}(\mathcal{O}_ X)$ to itself, see Simplicial, Lemma 14.34.2. Moreover, the augmentation

$\xymatrix{ f^* \ar[r] & f^*f_*f^* \ar@<1ex>[r] \ar@<-1ex>[r] & f^*f_*f^*f_*f^* \ar@<0ex>[l] \ar@<-2ex>[r] \ar@<0ex>[r] \ar@<2ex>[r] & f^*f_*f^*f_*f^*f_*f^* \ar@<1ex>[l] \ar@<-1ex>[l] }$

is a homotopy equivalence, see Simplicial, Lemma 14.34.3.

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. $\square$

Lemma 20.30.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}^\bullet$ be a bounded below complex of $\mathcal{O}_ X$-modules. There exists a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{G}^\bullet$ where $\mathcal{G}^\bullet$ be a bounded below complex of flasque $\mathcal{O}_ X$-modules and for all $x \in X$ the map $\mathcal{F}^\bullet _ x \to \mathcal{G}^\bullet _ x$ is a homotopy equivalence in the category of complexes of $\mathcal{O}_{X, x}$-modules.

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

$0 \to \mathcal{F}^\bullet \to f_*f^*\mathcal{F}^\bullet \to f_*f^*f_*f^*\mathcal{F}^\bullet \to f_*f^*f_*f^*f_*f^*\mathcal{F}^\bullet \to \ldots$

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

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

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

$\mathcal{F}^\bullet \to \mathcal{G}^\bullet = \text{Tot}( f_*f^*\mathcal{F}^\bullet \to f_*f^*f_*f^*\mathcal{F}^\bullet \to f_*f^*f_*f^*f_*f^*\mathcal{F}^\bullet \to \ldots )$

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. $\square$

Comment #6052 by Hans Schoutens on

In 2nd line of the statement of Lemma 0FKT, ...where F^\bullet be a bounded... should be ...where G^\bullet is a bounded...

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