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 pth 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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