# Dimension

My next goal is to work out the material of this post. To do this I am going prove that, given a separated morphisms f : X —> Y of algebraic spaces which is separated and locally of finite type, the functor which associates to T/Y the set of open subspaces Z \subset T \times_Y X which are finite over T is representable by an algebraic space. As a very first trivial step we will prove that the functor remains the same if we replace X by the open part of X where f has relative dimension 0…

But as happens frequently, we don’t have the prerequisites available. Namely, we have not yet discussed the relative dimension of morphisms of algebraic spaces in the stacks project. Thus the recent work on the stacks project is all about dimension of schemes, local rings, algebraic spaces, fibers of morphisms of algebraic spaces, etc. Everything seems to work exactly as expected — although we are not entirely done writing it all out — and given a morphism f : X —> Y of algebraic spaces which is locally of finite type, and an integer d there exists an open U_d of X which is exactly the set of points where f has relative dimension <= d.

PS: You can figure out what was added to the stacks project recently by clicking on the links under the heading “Development Logs” on the right hand side. The material in green is what was added and the material in red is the deleted lines. The commit messages sometimes give a brief indication of what is happening in the commit.