Lemma 59.84.3. Let $\Lambda $ be a Noetherian ring, let $k$ be an algebraically closed field, let $X$ be a separated finite type scheme over $k$ of dimension $\leq 1$, let $\mathcal{F}$ be a constructible sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$, and let $j : X' \to X$ be the inclusion of a dense open subscheme. Then $H^ q_{\acute{e}tale}(X, \mathcal{F})$ is a finite $\Lambda $-module if and only if $H^ q_{\acute{e}tale}(X, j_!j^{-1}\mathcal{F})$ is a finite $\Lambda $-module.
Proof. Since $X'$ is dense, we see that $Z = X \setminus X'$ has dimension $0$ and hence is a finite set $Z = \{ x_1, \ldots , x_ n\} $ of $k$-rational points. Consider the short exact sequence
\[ 0 \to j_!j^{-1}\mathcal{F} \to \mathcal{F} \to i_*i^{-1}\mathcal{F} \to 0 \]
of Lemma 59.70.8. Observe that $H^ q_{\acute{e}tale}(X, i_*i^{-1}\mathcal{F}) = H^ q_{\acute{e}tale}(Z, i^*\mathcal{F})$. Namely, $i : Z \to X$ is a closed immersion, hence finite, hence we have the vanishing of $R^ qi_*$ for $q > 0$ by Proposition 59.55.2, and hence the equality follows from the Leray spectral sequence (Cohomology on Sites, Lemma 21.14.6). Since $Z$ is a disjoint union of spectra of algebraically closed fields, we conclude that $H^ q_{\acute{e}tale}(Z, i^*\mathcal{F}) = 0$ for $q > 0$ and
\[ H^0_{\acute{e}tale}(Z, i^{-1}\mathcal{F}) = \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{F}_{x_ i} \]
which is a finite $\Lambda $-module $\mathcal{F}_{x_ i}$ is finite due to the assumption that $\mathcal{F}$ is a constructible sheaf of $\Lambda $-modules. The long exact cohomology sequence gives an exact sequence
\[ 0 \to H^0_{\acute{e}tale}(X, j_!j^{-1}\mathcal{F}) \to H^0_{\acute{e}tale}(X, \mathcal{F}) \to H^0_{\acute{e}tale}(Z, i^{-1}\mathcal{F}) \to H^1_{\acute{e}tale}(X, j_!j^{-1}\mathcal{F}) \to H^1_{\acute{e}tale}(X, \mathcal{F}) \to 0 \]
and isomorphisms $H^0_{\acute{e}tale}(X, j_!j^{-1}\mathcal{F}) \to H^0_{\acute{e}tale}(X, \mathcal{F})$ for $q > 1$. The lemma follows easily from this. $\square$
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Your email address will not be published.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)