The goal of this post is to introduce, in a very informal way, the notion of a reductive group, and discuss some examples.
This will be the first in a series of posts discussing Shimura varieties. In particular, we will focus here on a sort of broad motivation for the subject—why Shimura varieties are a natural thing to study and, in particular, what they give us.
In this post, I would just like to discuss a slightly different perspective on the étale cohomology of varieties. This might be called the ‘relative’ or ‘monodromy’ perspective, and it is rife with geometric intuition. While certainly first principles in some regards, it’s a point of view that I humbly believe is not emphasized well in most basic texts on the subject (e.g. Milne’s Lectures on Étale Cohomology).
In this post we will talk about the basic theory of group cohomology, including the cohomology of profinite groups.
We will assume that the reader is familiar with the basic theory of derived functors as in, say, Weibel’s Homological Algebra.
In this post we prove the well-known fact that for a smooth curve , the arithmetic genus agrees with the topological genus .