The Stacks project

Lemma 61.14.2. Let $X$ be a scheme. Let $E \in D^+(X_{pro\text{-}\acute{e}tale})$ be represented by a bounded below complex $\mathcal{E}^\bullet $ of abelian sheaves. Let $K$ be a hypercovering of $U \in \mathop{\mathrm{Ob}}\nolimits (X_{pro\text{-}\acute{e}tale})$ with $K_ n = \{ U_ n \to U\} $ where $U_ n$ is a weakly contractible object of $X_{pro\text{-}\acute{e}tale}$. Then

\[ R\Gamma (U, E) = \text{Tot}(s(\mathcal{E}^\bullet (K))) \]

in $D(\textit{Ab})$.

Proof. If $\mathcal{E}$ is an abelian sheaf on $X_{pro\text{-}\acute{e}tale}$, then the spectral sequence of Hypercoverings, Lemma 25.5.3 implies that

\[ R\Gamma (X_{pro\text{-}\acute{e}tale}, \mathcal{E}) = s(\mathcal{E}(K)) \]

because the higher cohomology groups of any sheaf over $U_ n$ vanish, see Cohomology on Sites, Lemma 21.51.1.

If $\mathcal{E}^\bullet $ is bounded below, then we can choose an injective resolution $\mathcal{E}^\bullet \to \mathcal{I}^\bullet $ and consider the map of complexes

\[ \text{Tot}(s(\mathcal{E}^\bullet (K))) \longrightarrow \text{Tot}(s(\mathcal{I}^\bullet (K))) \]

For every $n$ the map $\mathcal{E}^\bullet (U_ n) \to \mathcal{I}^\bullet (U_ n)$ is a quasi-isomorphism because taking sections over $U_ n$ is exact. Hence the displayed map is a quasi-isomorphism by one of the spectral sequences of Homology, Lemma 12.25.3. Using the result of the first paragraph we see that for every $p$ the complex $s(\mathcal{I}^ p(K))$ is acyclic in degrees $n > 0$ and computes $\mathcal{I}^ p(U)$ in degree $0$. Thus the other spectral sequence of Homology, Lemma 12.25.3 shows $\text{Tot}(s(\mathcal{I}^\bullet (K)))$ computes $R\Gamma (U, E) = \mathcal{I}^\bullet (U)$. $\square$

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