Lemma 69.6.6. Let $S$ be a scheme. Let $f : U \to X$ be a surjective, étale, and separated morphism of algebraic spaces over $S$. For $p \geq 0$ set
\[ W_ p = U \times _ X \ldots \times _ X U \setminus \text{all diagonals} \]
(with $p + 1$ factors) as in Lemma 69.6.4. Let $\chi _ p : S_{p + 1} \to \{ +1, -1\} $ be the sign character. Let $U_ p = W_ p/S_{p + 1}$ and $\underline{\mathbf{Z}}(\chi _ p)$ be as in Lemma 69.6.5. Then the spectral sequence of Lemma 69.6.3 has $E_1$-page
\[ E_1^{p, q} = H^ q(U_ p, \mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p)) \]
and converges to $H^{p + q}(X, \mathcal{F})$.
Proof.
Note that since the action of $S_{p + 1}$ on $W_ p$ is over $X$ we do obtain a morphism $U_ p \to X$. Since $W_ p \to X$ is étale and since $W_ p \to U_ p$ is surjective étale, it follows that also $U_ p \to X$ is étale, see Morphisms of Spaces, Lemma 67.39.2. Therefore an injective object of $\textit{Ab}(X_{\acute{e}tale})$ restricts to an injective object of $\textit{Ab}(U_{p, {\acute{e}tale}})$, see Cohomology on Sites, Lemma 21.7.1. Moreover, the functor $\mathcal{G} \mapsto \mathcal{G} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p))$ is an auto-equivalence of $\textit{Ab}(U_ p)$, whence transforms injective objects into injective objects and is exact (because $\underline{\mathbf{Z}}(\chi _ p)$ is an invertible $\underline{\mathbf{Z}}$-module). Thus given an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $ in $\textit{Ab}(X_{\acute{e}tale})$ the complex
\[ \Gamma (U_ p, \mathcal{I}^0|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p)) \to \Gamma (U_ p, \mathcal{I}^1|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p)) \to \Gamma (U_ p, \mathcal{I}^2|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p)) \to \ldots \]
computes $H^*(U_ p, \mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p))$. On the other hand, by Lemma 69.6.5 it is equal to the complex of $S_{p + 1}$-anti-invariants in
\[ \Gamma (W_ p, \mathcal{I}^0) \to \Gamma (W_ p, \mathcal{I}^1) \to \Gamma (W_ p, \mathcal{I}^2) \to \ldots \]
which by Lemma 69.6.4 is equal to the complex
\[ \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^0) \to \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^1) \to \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^2) \to \ldots \]
which computes $\mathop{\mathrm{Ext}}\nolimits ^*_{\textit{Ab}(X_{\acute{e}tale})}(K^ p, \mathcal{F})$. Putting everything together we win.
$\square$
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