Let $f:X \longrightarrow Y$ be a morphsim of Noetherian schemes. I was doing excersise 5.5 of Hartshorn Algebraic Geometry and in (c) i showed that finite morphisms preserve coherence (i.e. if $\mathscr{F}$ is coherent on $X$ then $f_*\mathscr{F}$ is coherent on $Y$).
Now I am wondering about something like a converse, suppose we have a morphism $f:X \longrightarrow Y$ of Noetherian schemes with $f_*\mathcal{O}_X$ a coherent $\mathcal{O}_Y$-module. What conditions de we need on $f$ for it to be finite? (I was thinking maybe about affine morphisms)
Or maybe, a more precise question: are affine morphsims with the condition that $f_*\mathcal{O}_X$ coherent, proper? That would work if it is the case.
Best Answer
Any proper morphism has the property that $f_*F$ is coherent if $F$ is. So, in particular, they need not be finite. Affine morphism with $f_*O_X$ coherent are proper. Putting these together, a morphism (of reasonable schemes) is finite if and only if it is proper and affine.