A projective $A$-scheme is a $\operatorname{Proj} S_{\bullet}$ where $S_{\bullet}$ is a finitely generated graded ring over $A=S_0$.
A quasi-projective $A$-scheme is an open quasicompact subscheme of a projective $A$-scheme.
An scheme over $A$ is a scheme where all its rings of sections are $A$-algebras and restriction maps are maps of $A$-algebras.
A scheme over A is called locally of finite type over $A$ if it can be covered by $\operatorname{Spec} A_i$ where the $A_i$’s are finitely generated $A$-algebras.
Now, Exercise 5.3.D of Vakil notes asks us to prove that quasi-projective A-schemes are locally of finite type over A.
Since the $D(f)\cong \operatorname{Spec} ((S_{\bullet})_f)_0)$ cover, I think that proving $((S_{\bullet})_f)_0)$ is a finitely generated $A$-algebra is enough, but I haven’t been capable to do it. Any help?