Lemma 30.7.3. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Assume that $f$ is quasi-compact and quasi-separated and that $S$ is quasi-compact and separated. There exists a bounded below complex $\mathcal{K}^\bullet $ of quasi-coherent $\mathcal{O}_ S$-modules with the following property: For every morphism $g : S' \to S$ the complex $g^*\mathcal{K}^\bullet $ is a representative for $Rf'_*\mathcal{F}'$ with notation as in diagram (30.5.0.1).
Proof. (If $f$ is separated as well, please see Lemma 30.7.2.) The assumptions imply in particular that $X$ is quasi-compact and quasi-separated as a scheme. Let $\mathcal{B}$ be the set of affine opens of $X$. By Hypercoverings, Lemma 25.11.4 we can find a hypercovering $K = (I, \{ U_ i\} )$ such that each $I_ n$ is finite and each $U_ i$ is an affine open of $X$. By Hypercoverings, Lemma 25.5.3 there is a spectral sequence with $E_2$-page
\[ E_2^{p, q} = \check{H}^ p(K, \underline{H}^ q(\mathcal{F})) \]
converging to $H^{p + q}(X, \mathcal{F})$. Note that $\check{H}^ p(K, \underline{H}^ q(\mathcal{F}))$ is the $p$th cohomology group of the complex
\[ \prod \nolimits _{i \in I_0} H^ q(U_ i, \mathcal{F}) \to \prod \nolimits _{i \in I_1} H^ q(U_ i, \mathcal{F}) \to \prod \nolimits _{i \in I_2} H^ q(U_ i, \mathcal{F}) \to \ldots \]
Since each $U_ i$ is affine we see that this is zero unless $q = 0$ in which case we obtain
\[ \prod \nolimits _{i \in I_0} \mathcal{F}(U_ i) \to \prod \nolimits _{i \in I_1} \mathcal{F}(U_ i) \to \prod \nolimits _{i \in I_2} \mathcal{F}(U_ i) \to \ldots \]
Thus we conclude that $R\Gamma (X, \mathcal{F})$ is computed by this complex.
For any $n$ and $i \in I_ n$ denote $f_ i : U_ i \to S$ the restriction of $f$ to $U_ i$. As $S$ is separated and $U_ i$ is affine this morphism is affine. Consider the complex of quasi-coherent sheaves
\[ \mathcal{K}^\bullet = ( \prod \nolimits _{i \in I_0} f_{i, *}\mathcal{F}|_{U_ i} \to \prod \nolimits _{i \in I_1} f_{i, *}\mathcal{F}|_{U_ i} \to \prod \nolimits _{i \in I_2} f_{i, *}\mathcal{F}|_{U_ i} \to \ldots ) \]
on $S$. As in Hypercoverings, Lemma 25.5.3 we obtain a map $\mathcal{K}^\bullet \to Rf_*\mathcal{F}$ in $D(\mathcal{O}_ S)$ by choosing an injective resolution of $\mathcal{F}$ (details omitted). Consider any affine scheme $V$ and a morphism $g : V \to S$. Then the base change $X_ V$ has a hypercovering $K_ V = (I, \{ U_{i, V}\} )$ obtained by base change. Moreover, $g^*f_{i, *}\mathcal{F} = f_{i, V, *}(g')^*\mathcal{F}|_{U_{i, V}}$. Thus the arguments above prove that $\Gamma (V, g^*\mathcal{K}^\bullet )$ computes $R\Gamma (X_ V, (g')^*\mathcal{F})$. This finishes the proof of the lemma as it suffices to prove the equality of complexes Zariski locally on $S'$. $\square$
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