Let $f:X \to Y$ a finite (especially affine) morphism between schemes and $F$ a coherent sheaf on $X$.
My question is how to verify that the pushforward $f_*F$ is also coherent?
Futhermore: Does this statement also holds for higher derived images $R^k f_*F$? Additionally can the assumption that $f$ is finite replaced by a weaker one (for example separated, proper or so on...)?
My ideas: Problem is finite and $f$ is affine, so wlog the problem translates to commutate algebra in following way:
$f: R \to A$ ring map and $M$ is $A$-module (this reflects the fact that $f_*F$ is quasi coherent).
Then $f_*M= M \vert _R$ is exactly the restriction of $M$ to $R$.
Does $f_*$ respect direct sums? (why?)
My hope would be to map a surjective map $A^r \to M$ (is given by coherence of $F = \overline{M}$) via $f_*$ to
$f_*A^r \to f_*M= M \vert _R$
So the problem depends on right exactness of $f_*$ and compatibility with sums. Could anybody help me to cope with these two obstructions?
Additionally, what would be the strategy for higher images $R^k f_*F$? These arise from presheaves $U \to H^k(U, f_*F)$? Since $f$ affine it seems that we also could work locally...