Let $\pi: X \to \text{Spec} A$ be a morphism. Is it true that $X$ is of finite type if $X$ can be covered by a finite affine cover $\text{Spec} B_i$ such that $A \to B_i$ gives $B_i$ structure of finitely generated $B$-algebra?
(My point is: instead of checking the above condition for every open affine of $\text{Spec} A$, does it suffice to just check it for $\text{Spec} A$ itself?)