Finiteness theorem in algebraic geometry

Hi,

let f: X -> Y a proper morphism of Noetherian schemes of finite type.
Then for any coherent O_X-module F the direct image f_*(F) and the
higher direct images R^qf_*(F), q>0, are all coherent O_Y-modules.

This is the finiteness theorem in the Noetherian setting. Its proof uses
the technique of devissage relying on Noetherian induction and Chow's lemma.

Two interesting issues which are not too obvious to me:

1. If the Noetherian assumption on X and Y is dropped and the
assumptions that f be separated and of finite presentation, Y be
quasi-compact, what versions of the finiteness theorems do exist?


2. If g: Y' -> Y is a base extension and the finiteness theorem is true
for the quasi-coherent O_X-module F of finite presentation (i.e.
f_*(F) is coherent), what assumptions on g are sufficient (other that
g being flat) that the module F x_Y Y' on the fiber product X x_Y
Y' has coherent direct image on Y'.

g can be an affine morphism here.

Thanks for your thoughts. References are highly appreciated.

--
Best wishes,
J.

.

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