Lemma 33.44.3 (0AYS)—The Stacks project

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Part 2: Schemes
Chapter 33: Varieties
Section 33.44: Degrees on curves
Lemma 33.44.3 (cite)

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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. $\square$

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