The Stacks project

Lemma 33.44.7. Let $k$ be a field, let $X$ be a proper scheme of dimension $\leq 1$ over $k$, and let $\mathcal{E}$, $\mathcal{V}$ be locally free $\mathcal{O}_ X$-modules of constant finite rank. Then

\[ \deg (\mathcal{E} \otimes \mathcal{V}) = \text{rank}(\mathcal{E}) \deg (\mathcal{V}) + \text{rank}(\mathcal{V}) \deg (\mathcal{E}) \]

Proof. By Lemma 33.44.6 and elementary arithmetic, we reduce to the case of a proper curve. This case follows from Lemma 33.44.5. $\square$

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