The Stacks project

Lemma 33.44.9. 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 33.33.2) and Lemma 33.33.3. $\square$

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