The Stacks project

Lemma 59.54.1. Let $f: X \to Y$ be a morphism and $\mathcal{I}$ an injective object of $\textit{Ab}(X_{\acute{e}tale})$. Let $V \in \mathop{\mathrm{Ob}}\nolimits (Y_{\acute{e}tale})$. Then

  1. for any covering $\mathcal{V} = \{ V_ j\to V\} _{j \in J}$ we have $\check H^ p(\mathcal{V}, f_*\mathcal{I}) = 0$ for all $p > 0$,

  2. $f_*\mathcal{I}$ is acyclic for the functor $\Gamma (V, -)$, and

  3. if $g : Y \to Z$, then $f_*\mathcal{I}$ is acyclic for $g_*$.

Proof. Observe that $\check{\mathcal{C}}^\bullet (\mathcal{V}, f_*\mathcal{I}) = \check{\mathcal{C}}^\bullet (\mathcal{V} \times _ Y X, \mathcal{I})$ which has vanishing higher cohomology groups by Lemma 59.18.7. This proves (1). The second statement follows as a sheaf which has vanishing higher Čech cohomology groups for any covering has vanishing higher cohomology groups. This a wonderful exercise in using the Čech-to-cohomology spectral sequence, but see Cohomology on Sites, Lemma 21.10.9 for details and a more precise and general statement. Part (3) is a consequence of (2) and the description of $R^ pg_*$ in Lemma 59.51.6. $\square$

Comments (0)

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Your email address will not be published.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).

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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03QB. Beware of the difference between the letter 'O' and the digit '0'.