# Is the pushforward via a proper map of a finite presentation module of finite presentation?

It's true that the pushforward of a coherent sheaf is coherent via a proper morphism: but do proper morphisms preserve a finite presentation? Under some assumptions perhaps? Does it change if we are working with algebraic spaces instead of just schemes?

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wait a minute: is the answer no, for trivial reasons? Take a non-finitely generated ideal $I$ in a ring $R$, then $R \to R/I$ wants to give a presentation of $R/I$ but it's not a finite presentation as $I$ is not finitely generated. –  Yosemite Sam Mar 28 '12 at 17:42
and we know that if $R/I$ admits a finite presentation then the kernel of any surjection $R^n \to R/I$ will be finitely generated. mathoverflow.net/questions/1788/… –  Yosemite Sam Mar 28 '12 at 17:43

Consider a ring $A$ and an ideal $I\subseteq A$, then $A/I$ is finitely presented as an $A/I$-module, but only finitely presented as an $A$-module if $I$ is finitely generated.