Finiteness of cohomology for pro-locally proper maps

We introduce a notion of proper morphism for schematic finite spaces and prove the analogue of Grothendieck’s finiteness theorem for it by means of the classic result for schemes and general descent arguments. This result also generalizes the class of morphisms of schemes for which the conclusion of the aforementioned Theorem holds. The key is giving a weaker definition of locally finitely presented morphisms.