Let $A \subset B$ be a finite extension of graded rings (so that $B$ is finite as a graded $A$-module). There is an induced finite morphism $\operatorname{Proj} B \to \operatorname{Proj} A$.
Is this morphism necessarily surjective?
Let $A \subset B$ be a finite extension of graded rings (so that $B$ is finite as a graded $A$-module). There is an induced finite morphism $\operatorname{Proj} B \to \operatorname{Proj} A$.
Is this morphism necessarily surjective?
If $\psi:A \to B$ is injective, then you can say that $\Psi : Proj(B) \to Proj(A)$ is dominant, that is has dense image.
Indeed, take an open set $D_+(f)\subset Proj(A)$ for some non-nilpotent graded element $f\in A$. We want to show that this open set intersects the image of $\Psi$.
To see this, we know that $\psi(f)$ is non nilpotent so that $D_+(\psi(f))$ is not empty. Therefore there exists a graded prime ideal $\mathfrak p \subset B$ such that $\mathfrak p \notin D_+(\psi(f))$.
Therefore $f \notin \psi^{-1}(\mathfrak p)$ and $\Psi(\mathfrak p)=\psi^{-1}(\mathfrak p) \in D_+(f)$, which concludes.
Remark. There is no need to assume finiteness for $\psi$.