In remark 5.16.1 the claim is made that if $i\colon X\to \mathbb{P}^r_Y$ is a closed immersion, then $i^*(\mathcal{O}_Y)$ is a very ample invertible sheaf on $X$, but I cannot determine why it is locally of rank 1. I tried to appeal directly to the definition of $i^*(\mathcal{O}(1))$, but wasn't able to make any sort of meaningful progress.
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1In the copy I'm looking at, he says in that remark that $i^* \mathcal O(1)$ is very ample which is purely definitional, not the sheaf you mentioned. – Tabes Bridges Sep 24 '20 at 21:40
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The pullback of the structure sheaf is the structure sheaf, which implies that the pullback of a locally free sheaf of rank $n$ is again a locally free sheaf of rank $n$. The linked duplicate tells you this. – KReiser Sep 24 '20 at 21:41