The Stacks project

Lemma 53.18.3. Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $f : Y \to X$ be a finite morphism such that there exists a dense open $U \subset X$ over which $f$ is a closed immersion. Then

\[ \dim _ k H^1(X, \mathcal{O}_ X) \geq \dim _ k H^1(Y, \mathcal{O}_ Y) \]

Proof. Consider the exact sequence

\[ 0 \to \mathcal{G} \to \mathcal{O}_ X \to f_*\mathcal{O}_ Y \to \mathcal{F} \to 0 \]

of coherent sheaves on $X$. By assumption $\mathcal{F}$ is supported in finitely many closed points and hence has vanishing higher cohomology (Varieties, Lemma 33.33.3). On the other hand, we have $H^2(X, \mathcal{G}) = 0$ by Cohomology, Proposition 20.20.7. It follows formally that the induced map $H^1(X, \mathcal{O}_ X) \to H^1(X, f_*\mathcal{O}_ Y)$ is surjective. Since $H^1(X, f_*\mathcal{O}_ Y) = H^1(Y, \mathcal{O}_ Y)$ (Cohomology of Schemes, Lemma 30.2.4) we conclude the lemma holds. $\square$

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Your email address will not be published.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 0CE3. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 53.18.3 as pdf