I’ve been reading a lot about descent theory lately, and I want to explain some of the ideas that I’ve absorbed, partially because I don’t fully understand them yet.
In algebraic geometry, we often like to glue things. In other words, we define something locally and have to “patch” the local things. An example is the of a quasi-coherent sheaf of algebras. Let be a graded quasi-coherent sheaf of algebras on the scheme . Then, for an open affine , is the sheaf associated to a graded -algebra . We can define the of this algebra; it is a scheme over . When we do this for each open affine and glue the resulting schemes , we get the of , which we can call . This is an example of how gluing is useful. Another example is the construction of the of a quasi-coherent sheaf of algebras. So gluing is ubiquitous.
We start with a review of the ideas behind gluing. Let’s now take the simplest possible example of how gluing might actually work in detail. Suppose we have a scheme and an open cover of , and quasi-coherent sheaves on for each . We would like to “glue ” the into one quasi-coherent sheaf on that restricts to each of the on each . In order to do this, we need isomorphisms
that satisfy the cocycle condition
Proposition 1 Given the above data, there is a unique quasi-coherent sheaf on with isomorphisms such that the isomorphisms correspond to the identity maps .
The idea is quite simple. Fix an open set . Then an element of is a collection of elements such that the restrictions of to map to each other under the isomorphisms . From the cocycle condition, it is easy to see that for . It is also easy to see that this indeed the sheaf in question, and since it is locally isomorphic to something quasi-coherent, it is itself quasi-coherent.
Descent theory is a kind of glorified gluing. It says that something similar to the above easy result holds when the notion of “cover” is replaced by something much weaker. Instead of having a bunch of open sets that cover , we might have a bunch of flat morphisms into whose images cover . Then, descent theory states that if we have quasi-coherent sheaves on which “glue” in a certain sense on (this is the analog of the intersections ), then we can reconstruct a sheaf on . When the are open immersions, then this is just the previous result. When they are only required to be flat and have images covering , it is much more profound.
It happens that there’s a bit more than faithful flatness that is actually needed; this is “quasi-compactness.” I will try to go into the details in the next series of posts. First, let us set up a framework of categorical nonsense that will be useful.
1. Grothendieck topologies
I mentioned that descent theory is a fancier version of gluing, where a “cover” by open sets is replaced by something more general. There is an axiomatic framework of this that works in categories.
So let be a category admitting arbitrary fibered products. We define a Grothendieck topology as follows.
Definition 2 A Grothendieck topology is an association that says whether a pair consisting of an object and a collection of arrows is considered a cover. This notion must satisfy the following axioms:
- Any isomorphism is a cover.
- If is a cover and is a morphism, then is a cover. Covers are stable under base-extension.
- If is a cover for each and is a cover, then is a cover of .
It is easy to see that a Grothendieck topology on the category of topological spaces and continuous maps can be defined by stipulating that a family of arrows is a cover if and only if each is an open immersion (i.e. isomorphism onto an open subset) and whose images cover . The axioms say, in English:
- The trivial cover is a cover, as we would expect; more generally, anything equivalent in this category to the trivial cover is a cover.
- Given a cover of , its inverse image under is a cover. Pull-backs are categorical version of inverse images.
- Covers are transitive.
As an example, here is an illustration of the axioms:
Proposition 3 Suppose is a family of morphisms which covers . Suppose is a family of isomorphisms. Then the family of covers .
Proof: Indeed, this follows because each is a cover of , and covers are transitive.
We can modify the previous example of a Grothendieck topology:
Example 1 Let be a fixed topological space and the category of open subsets of together with the inclusion maps. This has fibered products (which correspond simply to intersections). That is, , as is easily seen. Then we can define a Grothendieck topology on by saying that a bunch of arrows cover if and only if the images cover . In this case, a cover is literally an open cover.
We introduce one last bit of terminology before proceeding:
Definition 4 A category with a (Grothendieck) topology is called a site.
I’m following the terminology in FGA Explained, or alternatively Angelo Vistoli’s notes. I believe that SGA actually calls this a “pretopology.”
There are many interesting sites in algebraic geometry. Fix a scheme , and let be the category of schemes over .
where we have used a funny notation for the pre-image. Since the images of the cover , it follows that the are jointly surjective. This is a very fine topology, though. For instance, a scheme can be covered by a bunch of one-point schemes!
We will see that with a slight modification of the flat topology, the gluing procedure for quasi-coherent sheaves that I described above can still be done.