The Stacks project

Exercise 111.34.11. Let $A \to B$ be a finite type ring map. Suppose that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ factors through a closed immersion $\mathop{\mathrm{Spec}}(B) \to \mathbf{P}^ n_ A$ for some $n$. Prove that $A \to B$ is a finite ring map, i.e., that $B$ is finite as an $A$-module. Hint: if $A$ is Noetherian (please just assume this) you can argue using that $H^ i(Z, \mathcal{O}_ Z)$ for $i \in \mathbf{Z}$ is a finite $A$-module for every closed subscheme $Z \subset \mathbf{P}^ n_ A$.