These are lecture notes from a minicourse delivered at the Royal Institute of Technology (KTH), Stockholm, in 2008. The idea with the course was to focus solely on affine schemes to give a rapid path through some ideas of geometric invariant theory, with Luna's theorems as the basic goal. This means that a thorough discussion of stability issues and projectivity of quotients was sacrificed. I aimed at giving a preparation for reading Mumford's GIT, and at the same time covering interesting material that is not treated, or only very briefly, by Mumford.
The notes were written basically as handouts during the course. I intended to polish the text afterwards; of course I have not found the time for this. This means that proper citation and acknowledgements are not up to academic standards. For now I remedy this by stressing: everything in these notes is well known, and no results are my own. Among the sources I used most in the preparation of these lectures are:
Abstract: If a group scheme G acts on a scheme X, one is led to ask whether there exists a scheme deserving the name “the quotient X/G”. Mumford's geometric invariant theory addresses this problem for actions by reductive groups.I still hope to find an opportunity to clean up and expand these notes; in particular the last chapter needs more work.
In this course we will focus on the affine case X = Spec(A). The quotient X/G is then the spectrum Spec(AG) of the invariant ring. It is usually hard to understand the quotient (e.g. is it smooth?) by analysing the invariant ring directly. The central result that enables us to answer many local questions is Luna's etale slice theorem. In particular, Luna's theorem can be used to show that a quotient X → X/G is a principal G-bundle if and only if all stabilizers are trivial.
I will give a reasonably self-contained presentation, beginning with basic concepts surrounding group schemes and their actions, and ending up with Luna's theorem and its applications.