The semester is completely over here at Columbia University, so I have more time to work on the stacks project. Since the last update (May 14) we have made the following changes to the stacks project
- Moved the more technical and advanced material on groupoid schemes to its own chapter, with the unimagitive title “More on Groupoid Schemes”.
- Rewrote some lemmas on local properties of groupoids for greater clarity, and to make them more widely applicable.
- Small reorganization of the material on quotient stacks.
- Added the following result to the chapter on varieties: If X is a variety over an algebraically closed field k then O(X)^*/k^* is a finitely generated abelian group. Somehow we will need this result in the near future.
- This forced us to rethink some of the material on geometrically irreducible/reduced/connected schemes over fields. Leading to a bunch of small improvements.
- Fixed a circular reasoning in the algebra chapter.
- Finally, we added some stuff on groupoids on fields which we discuss below.
Of course this is a bit boring but I wanted to show that in the course of working towards a new result in the stacks project there is a kind of tendency to explain earlier material better and more precisely. In fact, in this manner we go over most of the material in the stacks project multiple times, and the material that gets used more is looked at more often — hopefully leading to an ever improved version of the most used algebraic geometry results in the stacks project.
A “groupoid on a field” means a groupoid scheme (U, R, s, t, c) where U is the spectrum of a field. These are quite interesting objects to work with, somehow analogous to group schemes over fields. For example here are some results we have added to the stacks project so far:
- c : R \times_{s, U, t} R \to R is open,
- R is a separated scheme,
- there is a unique irreducible component Z of R which passes through the identity e,
- Z is geometrically irreducible via both s and t,
- if s, t are locally of finite type, then R is equidimensional.
We intend to add a few more soon. The main target is to show that if s,t are locally of finite type, then dim(R) = dim(G) where G is the stabilizer group scheme, and the material above goes into the details of the approach to this I have in mind. But I wonder if there is some completely general theorem saying that a groupoid on a field is somehow an extension of a group acting on a field by a group scheme over the field (how to formulate this precisely is not completely clear to me). Ideas?