Lemma 75.5.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Then the map $E \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}E$ of Derived Categories, Remark 13.34.4 is an isomorphism1.
Proof. Denote $\mathcal{H}^ i = H^ i(E)$ the $i$th cohomology sheaf of $E$. Let $\mathcal{B}$ be the set of affine objects of $X_{\acute{e}tale}$. Then $H^ p(U, \mathcal{H}^ i) = 0$ for all $p > 0$, all $i \in \mathbf{Z}$, and all $U \in \mathcal{B}$ as $U$ is an affine scheme. See discussion in Cohomology of Spaces, Section 69.3 and Cohomology of Schemes, Lemma 30.2.2. Thus the lemma follows from Cohomology on Sites, Lemma 21.23.10 with $d = 0$. $\square$
Post a comment
Your email address will not be published.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)