Let $f:X \to Y$ a proper, affine morphism between Noetherian schemes $X,Y$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. It is well known result that for all $r \ge 0$ the higher direct image $R^r f_*\mathcal{F}$ is a coherent $\mathcal{O}_Y$-module.
The roughly idea is to split the proof in several steps:
- reduce to affine case $Y=Spec(R)$
- case $f$ structure map $\mathbb{P}^n_Y \to Y$
- case $f$ projective
- general case: use Chow's lemma, Grothendieck spectral sequence and noetherian induction
Question: Is it possible to proof the claim in the case $\mathcal{F}=\mathcal{O}_X$ and $r=0$ with more elementary methods; preferably without Chow's lemma and spectral theory, i.e. to show that direct image $f_* \mathcal{O}_X$ is coherent $\mathcal{O}_Y$-module?
by 'elementary' I refer to usual methods from commutative algebra and ring theory. I proceed as follows: $f$ affine and thus as in step 1. I assume $Y=Spec(R), X=Spec(A)$ affine and the induced ring homomorphism by $f$ is $\phi: R \to A$ and our goal is to show that $A$ is coherent $R$-module. This means that $A$ is a quotient of free module $R^n$ for appropriate $n >0$.
As $f$ proper, by definition $A$ is a finitely generated $R$-algebra: this guarantees the existence of a surjection $R[X_1,...,X_n] \to A$ of $R$-algebras. But what we need is that $A$ is finitely generated as $R$-module.