Varieties

The most basic class of geometric object encountered in algebraic geometry is of course varieties. Before talking about cohomology of varieties, it seems sensible to say a few words about different types of varieties. I will assume that you know what a variety is – if not, look at Hartshorne or any other introductory book on algebraic geometry, or online notes of Dolgachev, Milne, Debarre, Vakil, Gathmann and other people.

A cohomology theory for varieties will typically be a functor from some category of varieties to the category of abelian groups or vector spaces. When reading about cohomology for some class of varieties, there are three key questions to ask about the category of varieties considered. 

Question 1: Are the varieties required to be complete/proper/projective? Although these words don’t mean exactly the same thing, they are morally and for most practical purposes the same. 

Question 2: Are the varieties required to be smooth/nonsingular?

Question 3: What is the base field?

The possible answers to Q1 and Q2 give us four possible classes of varieties: Smooth proper varieties, smooth varieties not necessarily proper, proper varieties not necessarily smooth, and general varieties. The third of these seems to be less common, so excluding it leaves us with the three most important classes of varieties, in increasing complexity:

• A: Smooth proper varieties (really nice and well-behaved)
• B: Smooth varieties (a bit more complicated, but still nice)
• C: Arbitrary varieties (nasty things, very hard to understand)

When seeing a category of varieties being introduced, it is often useful to make an internal note of which of the three situations we are in. For example, someone talking about “quasiprojective nonsingular varieties” would be in class B, someone talking about “smooth projective varieties” would be in class A, and someone talking about “integral separated schemes of finite type over the base field” is in class C. In most texts, the author states in the very beginning what he means by “variety”, and it is often one of the first two. We will see later that the right notion of cohomology depends on which situation we are in.

Question 3 also has a big impact on the study of cohomology theories for the varieties in question. Different cohomology theories are defined for different base fields. The most common base fields are: Finite fields, global fields and local fields, algebraic closures of these fields, and the fields and .

We will soon start looking at cohomology theories for smooth projective varieties, i.e. Weil cohomology theories.