The previous series on algebraic groups is over. Actually, I barely got to the root system and root datum of a reductive group, but I found that the whole slew of material was getting too complex to organise on this blog, which I feel is better for more self-contained posts. Instead, I have begun to write a set of notes in real LaTeX describing the root datum of a reductive group and root systems in general, most of which I have completed, and I will release a draft soon.

In the mean time, I will continue posts on algebraic groups by reviewing some theorem and then illustrating it with an application such an an exercises from some textbook. Let's start by reviewing a simple one: let $ k$ be an algebraically closed field. Then an irreducible finite set in the Zariski topology on $ k^n$ (as a classical variety) necessarily has one element. (A topological space is irreducible if it cannot be written as the union of two proper closed subsets.)

We added the disclaimer "as a classical variety" since over an algebraically closed field $ k$ it is sufficient to consider the closed points of $ \mathrm{Spec}(k[x_i])$—otherwise the statement would be false.

More »