Loading web-font TeX/Caligraphic/Regular

The Stacks project

Lemma 21.24.1. In the situation described above. Denote \mathcal{H}^ m = H^ m(\mathcal{F}^\bullet ) the mth cohomology sheaf. Let \mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) be a subset. Let d \in \mathbf{N}. Assume

  1. every object of \mathcal{C} has a covering whose members are elements of \mathcal{B},

  2. for every U \in \mathcal{B} we have H^ p(U, \mathcal{H}^ q) = 0 for p > d and q < 01.

Then (21.24.0.1) is a quasi-isomorphism.

Proof. By Derived Categories, Lemma 13.34.5 it suffices to show that the map \mathcal{F}^\bullet \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} \mathcal{F}^\bullet is an isomorphism. This follows from Lemma 21.23.10. \square

[1] It suffices if \forall m, \exists p(m), H^ p(U. \mathcal{H}^{m - p}) = 0 for p > p(m), see Lemma 21.23.9.

Comments (0)


Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.