Lemma 69.6.3. Let S be a scheme. Let f : U \to X be a surjective étale morphism of algebraic spaces over S. Let \mathcal{F} be an object of \textit{Ab}(X_{\acute{e}tale}). There exists a canonical map
\check{\mathcal{C}}^\bullet _{alt}(f, \mathcal{F}) \longrightarrow R\Gamma (X, \mathcal{F})
in D(\textit{Ab}). Moreover, there is a spectral sequence with E_1-page
E_1^{p, q} = \mathop{\mathrm{Ext}}\nolimits _{\textit{Ab}(X_{\acute{e}tale})}^ q(K^ p, \mathcal{F})
converging to H^{p + q}(X, \mathcal{F}) where K^ p = \wedge ^{p + 1}f_!\underline{\mathbf{Z}}.
Proof.
Recall that we have the quasi-isomorphism K^\bullet \to \underline{\mathbf{Z}}[0], see (69.6.1.1). Choose an injective resolution \mathcal{F} \to \mathcal{I}^\bullet in \textit{Ab}(X_{\acute{e}tale}). Consider the double complex \mathop{\mathrm{Hom}}\nolimits (K^\bullet , \mathcal{I}^\bullet ) with terms \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^ q). The differential d_1^{p, q} : A^{p, q} \to A^{p + 1, q} is the one coming from the differential K^{p + 1} \to K^ p and the differential d_2^{p, q} : A^{p, q} \to A^{p, q + 1} is the one coming from the differential \mathcal{I}^ q \to \mathcal{I}^{q + 1}. Denote \text{Tot}(\mathop{\mathrm{Hom}}\nolimits (K^\bullet , \mathcal{I}^\bullet )) the associated total complex, see Homology, Section 12.18. We will use the two spectral sequences ({}'E_ r, {}'d_ r) and ({}''E_ r, {}''d_ r) associated to this double complex, see Homology, Section 12.25.
Because K^\bullet is a resolution of \underline{\mathbf{Z}} we see that the complexes
\mathop{\mathrm{Hom}}\nolimits (K^\bullet , \mathcal{I}^ q) : \mathop{\mathrm{Hom}}\nolimits (K^0, \mathcal{I}^ q) \to \mathop{\mathrm{Hom}}\nolimits (K^1, \mathcal{I}^ q) \to \mathop{\mathrm{Hom}}\nolimits (K^2, \mathcal{I}^ q) \to \ldots
are acyclic in positive degrees and have H^0 equal to \Gamma (X, \mathcal{I}^ q). Hence by Homology, Lemma 12.25.4 the natural map
\mathcal{I}^\bullet (X) \longrightarrow \text{Tot}(\mathop{\mathrm{Hom}}\nolimits (K^\bullet , \mathcal{I}^\bullet ))
is a quasi-isomorphism of complexes of abelian groups. In particular we conclude that H^ n(\text{Tot}(\mathop{\mathrm{Hom}}\nolimits (K^\bullet , \mathcal{I}^\bullet ))) = H^ n(X, \mathcal{F}).
The map \check{\mathcal{C}}^\bullet _{alt}(f, \mathcal{F}) \to R\Gamma (X, \mathcal{F}) of the lemma is the composition of \check{\mathcal{C}}^\bullet _{alt}(f, \mathcal{F}) \to \text{Tot}(\mathop{\mathrm{Hom}}\nolimits (K^\bullet , \mathcal{I}^\bullet )) with the inverse of the displayed quasi-isomorphism.
Finally, consider the spectral sequence ({}'E_ r, {}'d_ r). We have
E_1^{p, q} = q\text{th cohomology of } \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^0) \to \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^1) \to \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^2) \to \ldots
This proves the lemma.
\square
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