Affine Quotients of Algebraic Groups
I will discuss a short paper of Borel called "On affine algebraic homogeneous spaces." He addresses the following question: given a connected affine algebraic group G over a field k and a subgroup H, when is G/H affine? It is well-known that when H is a Borel subgroup or more generally a parabolic subgroup, then G/H is a projective variety. For example, if G = GL_n and H is the upper triangular Borel, then G/H is isomorphic to the full flag variety. On the other end of the spectrum, if H is normal, then G/H is an affine algebraic group. However, G usually has lots of other subgroups, where it is not a priori clear what the quotient should be.
I will review the relevant concepts from algebraic groups and then sketch the proof of Borel's answer to this question, which is a nice application of some fairly elementary etale cohomology.