Lemma 33.43.3. 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 33.32.2) and additivity of ranks.
$\square$

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