  We prove that cohomology and base change holds for algebraic stacks,
generalizing work of Brochard in the tame case. We also show that Hom-spaces on
algebraic stacks are represented by abelian cones, generalizing results of
Grothendieck, Brochard, Olsson, Lieblich, and Roth--Starr. To accomplish all of
this, we prove that a wide class of Ext-functors in algebraic geometry are
coherent (in the sense of M. Auslander).
