Source: Coherent locally free sheaves and projective modules

Question

What is a good and very quick and concise article for the proof of the equivalence of the categories of locally free sheaves on $\mathrm{Spec}(A)$ and finitely generated projective $A$-modules?

Answer 1

Chapter II, §5.2 in:

N. Bourbaki, Elements of mathematics. Commutative algebra. Hermann, Paris, 1972. Translated from the French.

Alternatively, you can always consult the stacks project for these basics. In this case, it's Tag 00NV.

Answer 2

Why not read Serre's crystal-clear article which introduced this equivalence?
As Gauss said: "Read the masters".