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If $X$ is a noetherian scheme and $\mathcal{F}$ is a coherent locally free sheaf on $X$, we define the rank of $\mathcal{F}$ over some trivialising open subset $U \subseteq X$ to be the smallest integer $n$ such that $$ \mathcal{F}|_{U} \simeq \mathcal{O}_{X}|_{U}^{\oplus n}. $$ We can define a rank function via Nakayama's lemma and show that if $X$ is connected then there is a globally well-defined rank. This is all standard. My confusion is in Hartshorne, II Proposition 7.11. There he simply gives us that $X$ is a noetherian scheme, $\mathscr{E}$ is coherent locally free, and that $\text{rank }\mathscr{E} \geq 2$. Is there some more general definition of rank that would make this well-defined for an arbitrary noetherian scheme $X$? Or does he mean the largest rank over each of the connected components? I have never seen rank used this way before, so I'm really not sure.

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Here you can interpret $\text{rank }\mathscr{E} \geq 2$ as meaning that $\mathscr{E}$ has rank at least $2$ on each connected component. As far as I know this isn't exactly a standard usage of "rank", but it is the most sensible interpretation in context and it is what you need for the result to be true. Indeed, the proof is by looking locally to reduce to the case where $\mathscr{E}$ is free, and so you just need that it is always free of rank at least $2$ locally. Globally, that just means its rank on each connected component is at least $2$.

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