Lemma 33.44.3. Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $0 \to \mathcal{E}_1 \to \mathcal{E}_2 \to \mathcal{E}_3 \to 0$ be a short exact sequence of locally free $\mathcal{O}_ X$-modules each of finite constant rank. Then

\[ \deg (\mathcal{E}_2) = \deg (\mathcal{E}_1) + \deg (\mathcal{E}_3) \]

**Proof.**
Follows immediately from additivity of Euler characteristics (Lemma 33.33.2) and additivity of ranks.
$\square$

## 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Your email address will not be published.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).

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.

## Comments (0)

There are also: