Is there a natural hypothesis that one can put on a finite type morphism $f:X \to Y$ (say $Y$ is locally Noetherian) so that the direct image $f_*\mathcal{O}_X$ is a sheaf of finitely generated $\mathcal O_Y$-algebras?
Of cours, if $f$ is affine, or proper it is the case, but these two hypotheses seem trivial thing or an overkill, respectively.
Are there any other ones?
Of course one has to ask for something, since there are varieties over any field whose ring of sections is not f.g., but I cannot come up with any idea.
