Suppose we have a space . There are a lot of ways to describe the cohomology of — algebraic geometers will know about étale cohomology, crystalline cohomology and (algebraic) deRham cohomology; topologists would add singular cohomology and simplicial cohomology to the list. Recently, thanks in large part to MIT’s K-theory seminar, I’ve come to understand how to tie the first three together. Each cohomology theory comes from a different way of looking at local systems.
Roughly, a local system consists of a vector bundle on , together with some additional data which gives isomorphisms between different fibers of . In algebraic geometry, there are many different ways to make this idea precise.
In this series of posts, I want to present several of these ideas in the case of a smooth manifold . In each case, if you pursued the idea far enough, you would get to a major tool of modern algebraic geometry.
I said above that a local system includes the data of isomorphisms between different fibers and of . This data can depend on the choice of a path between and . Our three theories will be distinguished by how short the path is required to be.
We could work with an arbitrary . This would lead to étale sheaves
We could work with an infinitesimal (also known as an -jet). This would lead to crystals
We could work with a which is so small that it becomes a tangent vector. This would lead to -modules
These give rise to étale cohomology, crystalline cohomology and deRham cohomology. I should point out that, for modern applications, one usually wants to work on stacks, with singularities and in arbitrary characteristic. I won’t be addressing any of those issues; I just want to give the intuition behind each theory.
In this post, I will explain how to build a cohomology theory, given a notion of local system. I will then follow up with three more posts, one for each of the specific approaches above.
As usual, this series comes with a disclaimer: These are tools that are relevant to my work, but their inner workings are not my expertise. If you want to see experts discussing this sort of thing, it looks like that conversation is going on at Urs’ journal club.
How to build a cohomology theory
Let’s suppose we have something that acts like a category of local systems. In this category, there will be a trivial object . This will consist of the trivial line bundle over and, for any two points and , the trivial isomorphism between the fibers and .
Also, there will be a functor from local systems to vector spaces. For any local system , with underlying vector bundle , will be the vector space of sections of such that the isomorphism(s) between and take to .
In order to build a cohomology theory, you must find a way to embed your category into an abelian category (or a triangulated one). Then you will be able to talk about derived functors. The cohomology groups of will be . Those who are used to computing with derived functors will know that it is very important, therefore, to find explicit resolutions of in our various categories. This task will, roughly speaking, become easier as we move down our list of theories.
This sounds discussion probably sounds very abstract; one of the reasons that the different perspectives above are useful is that they provide different ways to make this concrete.
Many of the difficulties in these subjects come about in constructing the larger abelian category. This is another important issue that I will gesture at, but not address.