Finiteness of Cohomology for Pro-locally Proper Maps

Abstract We introduce a notion of proper morphism for schematic finite spaces and prove the analogue of Grothendieck's finiteness theorem for it. As a corollary, this result generalizes the class of morphisms of schemes for which the conclusion of the aforementioned Theorem holds. The techniques we employ, which further develop the theory of schematic spaces and proschemes, are ultimately founded on descent properties of flat epimorphisms of rings that are applicable in other situations in order to weaken finite presentation requirements.