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
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
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
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 $R\Gamma (U, E) = \mathcal{I}^\bullet (U)$. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)