# ag.algebraic geometry – Relative affine schemes

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.