The Stacks project

Lemma 33.43.9. Let $k$ be a field, let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $D$ be an effective Cartier divisor on $X$. Then $D$ is finite over $\mathop{\mathrm{Spec}}(k)$ of degree $\deg (D) = \dim _ k \Gamma (D, \mathcal{O}_ D)$. For a locally free sheaf $\mathcal{E}$ of rank $n$ we have

\[ \deg (\mathcal{E}(D)) = n\deg (D) + \deg (\mathcal{E}) \]

where $\mathcal{E}(D) = \mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(D)$.

Proof. Since $D$ is nowhere dense in $X$ (Divisors, Lemma 31.13.4) we see that $\dim (D) \leq 0$. Hence $D$ is finite over $k$ by Lemma 33.20.2. Since $k$ is a field, the morphism $D \to \mathop{\mathrm{Spec}}(k)$ is finite locally free and hence has a degree (Morphisms, Definition 29.48.1), which is clearly equal to $\dim _ k \Gamma (D, \mathcal{O}_ D)$ as stated in the lemma. By Divisors, Definition 31.14.1 there is a short exact sequence

\[ 0 \to \mathcal{O}_ X \to \mathcal{O}_ X(D) \to i_*i^*\mathcal{O}_ X(D) \to 0 \]

where $i : D \to X$ is the closed immersion. Tensoring with $\mathcal{E}$ we obtain a short exact sequence

\[ 0 \to \mathcal{E} \to \mathcal{E}(D) \to i_*i^*\mathcal{E}(D) \to 0 \]

The equation of the lemma follows from additivity of the Euler characteristic (Lemma 33.32.2) and Lemma 33.32.3. $\square$


Comments (0)

There are also:

  • 4 comment(s) on Section 33.43: Degrees on curves

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Your email address will not be published.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