Today I wrote a bit about the finite part of a morphism. The goal is to show: If f : X —> Y is locally of finite type and separated then the functor (X/Y)_{fin} which associates to a scheme T the set

{(a, Z) where a : T —> Y is a map and Z ⊂ T x_Y X is open and finite over T}

is representable by an algebraic space. It is easy to prove that it is a sheaf for the fppf topology. What is very cute is that it is trivial to show that (X/Y)_{fin} has representable diagonal. Hence now the only thing left to prove is that it has a surjective etale covering by a scheme which I think I know how to do.

As I expected this is quite a bit easier than proving representability theorems for Hilbert functors, which is the other method to approach the current short term goal: etale splitting of groupoids.