Q. Why a structure sheaf is a vector bundle?
For example, in the following youtube, in 4:27, Takuro Mochizuki uses a strucure sheaf $\mathcal{O}_X$ as an example of vector bundle; $E=\mathcal{O}_X$. https://www.youtube.com/watch?v=IuTIgdH7n1k
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2It “is” (= corresponds to) the trivial vector bundle of rank 1. In general we think of locally free $O_X$-modules of finite rank as vector bundles. – Zhen Lin Aug 31 '20 at 00:51
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1This has been covered extensively here before. – KReiser Aug 31 '20 at 01:44