Lemma 17.14.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. Suppose that the support of $\mathcal{O}_ X$ is $X$, i.e., all stalks of $\mathcal{O}_ X$ are nonzero rings. Let $\mathcal{F}$ be a locally free sheaf of $\mathcal{O}_ X$-modules. There exists a locally constant function

$\text{rank}_\mathcal {F} : X \longrightarrow \{ 0, 1, 2, \ldots \} \cup \{ \infty \}$

such that for any point $x \in X$ the cardinality of any set $I$ such that $\mathcal{F}$ is isomorphic to $\bigoplus _{i\in I} \mathcal{O}_ X$ in a neighbourhood of $x$ is $\text{rank}_\mathcal {F}(x)$.

Proof. Under the assumption of the lemma the cardinality of $I$ can be read off from the rank of the free module $\mathcal{F}_ x$ over the nonzero ring $\mathcal{O}_{X, x}$, and it is constant in a neighbourhood of $x$. $\square$

Comment #6818 by Yuto Masamura on

It may be good to note that $\operatorname{rank}_{\mathcal F}(x)$ is also denoted by $\operatorname{rank}_x(\mathcal F)$, for exmple see Lemma 29.34.12.

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