I was reading these notes by D. Gaitsgory, and I don’t understand a claim he makes about relative affine schemes. Namely, on page 3 he says that if $f: Y rightarrow X$ is an affine scheme over $X$, then there exist two vector bundles $E_1$, $E_2$ over $X$ together with a map $mathrm{Tot}(E_1) rightarrow mathrm{Tot}(E_2)$ such that $Y simeq mathrm{Tot}(E_1) times_{mathrm{Tot}(E_2)} X$, where $X subset mathrm{Tot}(E_2)$ is the zero section. Here $X$ is projective and $f$ is quasi-projective.

I don’t understand why this description exists. I agree that locally this is true (simply because $f$ is of finite type, hence we have that $f_{ast} mathcal{O}_Y$ is locally a finitely generated algebra – I am assuming $X$ is Noetherian), but I don’t see why one should be able to glue the local presentations to a global one.

Thanks for the help.