Lemma 33.44.5. Let $k$ be a field. Let $X$ be a proper curve over $k$ with generic point $\xi $. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $n$ and let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then

\[ \chi (X, \mathcal{E} \otimes \mathcal{F}) = r \deg (\mathcal{E}) + n \chi (X, \mathcal{F}) \]

where $r = \dim _{\kappa (\xi )} \mathcal{F}_\xi $ is the rank of $\mathcal{F}$.

**Proof.**
Let $\mathcal{P}$ be the property of coherent sheaves $\mathcal{F}$ on $X$ expressing that the formula of the lemma holds. We claim that the assumptions (1) and (2) of Cohomology of Schemes, Lemma 30.12.6 hold for $\mathcal{P}$. Namely, (1) holds because the Euler characteristic and the rank $r$ are additive in short exact sequences of coherent sheaves. And (2) holds too: If $Z = X$ then we may take $\mathcal{G} = \mathcal{O}_ X$ and $\mathcal{P}(\mathcal{O}_ X)$ is true by the definition of degree. If $i : Z \to X$ is the inclusion of a closed point we may take $\mathcal{G} = i_*\mathcal{O}_ Z$ and $\mathcal{P}$ holds by Lemma 33.33.3 and the fact that $r = 0$ in this case.
$\square$

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