The Stacks project

Lemma 68.6.6. Let $S$ be a scheme. Let $f : U \to X$ be a surjective, étale, and separated morphism of algebraic spaces over $S$. For $p \geq 0$ set

\[ W_ p = U \times _ X \ldots \times _ X U \setminus \text{all diagonals} \]

(with $p + 1$ factors) as in Lemma 68.6.4. Let $\chi _ p : S_{p + 1} \to \{ +1, -1\} $ be the sign character. Let $U_ p = W_ p/S_{p + 1}$ and $\underline{\mathbf{Z}}(\chi _ p)$ be as in Lemma 68.6.5. Then the spectral sequence of Lemma 68.6.3 has $E_1$-page

\[ E_1^{p, q} = H^ q(U_ p, \mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p)) \]

and converges to $H^{p + q}(X, \mathcal{F})$.

Proof. Note that since the action of $S_{p + 1}$ on $W_ p$ is over $X$ we do obtain a morphism $U_ p \to X$. Since $W_ p \to X$ is étale and since $W_ p \to U_ p$ is surjective étale, it follows that also $U_ p \to X$ is étale, see Morphisms of Spaces, Lemma 66.39.2. Therefore an injective object of $\textit{Ab}(X_{\acute{e}tale})$ restricts to an injective object of $\textit{Ab}(U_{p, {\acute{e}tale}})$, see Cohomology on Sites, Lemma 21.7.1. Moreover, the functor $\mathcal{G} \mapsto \mathcal{G} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p))$ is an auto-equivalence of $\textit{Ab}(U_ p)$, whence transforms injective objects into injective objects and is exact (because $\underline{\mathbf{Z}}(\chi _ p)$ is an invertible $\underline{\mathbf{Z}}$-module). Thus given an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $ in $\textit{Ab}(X_{\acute{e}tale})$ the complex

\[ \Gamma (U_ p, \mathcal{I}^0|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p)) \to \Gamma (U_ p, \mathcal{I}^1|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p)) \to \Gamma (U_ p, \mathcal{I}^2|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p)) \to \ldots \]

computes $H^*(U_ p, \mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p))$. On the other hand, by Lemma 68.6.5 it is equal to the complex of $S_{p + 1}$-anti-invariants in

\[ \Gamma (W_ p, \mathcal{I}^0) \to \Gamma (W_ p, \mathcal{I}^1) \to \Gamma (W_ p, \mathcal{I}^2) \to \ldots \]

which by Lemma 68.6.4 is equal to the complex

\[ \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^0) \to \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^1) \to \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^2) \to \ldots \]

which computes $\mathop{\mathrm{Ext}}\nolimits ^*_{\textit{Ab}(X_{\acute{e}tale})}(K^ p, \mathcal{F})$. Putting everything together we win. $\square$


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 0728. Beware of the difference between the letter 'O' and the digit '0'.