Lemma 31.12.5. Let $X$ be an integral scheme. Let $\mathcal{F}$ and $\mathcal{G}$ be quasi-coherent $\mathcal{O}_ X$-modules.
The rank of $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$ is the product of the ranks of $\mathcal{F}$ and $\mathcal{G}$.
If $\mathcal{F}$ is of finite presentation, then the rank of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is the product of the ranks of $\mathcal{F}$ and $\mathcal{G}$.
Proof. Omitted. Note that part (2) makes sense because $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is quasi-coherent by item (10) in Schemes, Section 26.24. The algebraic version of this lemma is More on Algebra, Lemma 15.23.4. $\square$
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