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 {\mathrm{proj}} of a quasi-coherent sheaf of algebras. Let {\mathcal{A}} be a graded quasi-coherent sheaf of algebras on the scheme {X}. Then, for an open affine {U = \mathrm{Spec} A}, {\mathcal{A}|_U} is the sheaf associated to a graded {A}-algebra {\Gamma(U, \mathcal{A})}. We can define the {\mathrm{Proj} } of this algebra; it is a scheme {X_U} over {U}. When we do this for each {U} open affine and glue the resulting schemes {X_U}, we get the {\mathrm{Proj}} of {\mathcal{A}}, which we can call {X}. This is an example of how gluing is useful. Another example is the construction of the {\mathrm{Spec}} 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 {X} and an open cover {\left\{U_i\right\}} of {X}, and quasi-coherent sheaves {\mathcal{F}_i} on {U_i} for each {i}. We would like to “glue ” the {\mathcal{F}_i} into one quasi-coherent sheaf on {X} that restricts to each of the {\mathcal{F}_i} on each {U_i}. In order to do this, we need isomorphisms

\displaystyle  \phi_{ij}: \mathcal{F}_i|_{U_i \cap U_j} \rightarrow \mathcal{F_j}|_{U_j \cap U_i}

that satisfy the cocycle condition

\displaystyle \phi_{jk } \circ \phi_{ij} = \phi_{ik}: \mathcal{F_i}|_{U_i \cap U_j \cap U_k} \rightarrow \mathcal{F_k}|_{U_i \cap U_j \cap U_k}.

Proposition 1 Given the above data, there is a unique quasi-coherent sheaf {\mathcal{F}} on {X} with isomorphisms {\mathcal{F}|_{U_i} \simeq \mathcal{F}_i} such that the isomorphisms {\phi_{ij}} correspond to the identity maps {\mathcal{F}|_{U_i \cap U_j} \rightarrow \mathcal{F}|_{U_i \cap U_j}}.

The idea is quite simple. Fix an open set {U \subset X}. Then an element of {\mathcal{F}(U)} is a collection of elements {a_i \in \mathcal{F}_i(U \cap U_i)} such that the restrictions of {a_i, a_j} to {\mathcal{F}_i(U \cap U_i \cap U_j), \mathcal{F}_j(U \cap U_i \cap U_j)} map to each other under the isomorphisms {\phi_{ij}}. From the cocycle condition, it is easy to see that {\mathcal{F}(V)\simeq \mathcal{F}_i(V)} for {V \subset U_i}. 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 {X}, we might have a bunch of flat morphisms {X_i \rightarrow X} into {X} whose images cover {X}. Then, descent theory states that if we have quasi-coherent sheaves on {X_i} which “glue” in a certain sense on {X_i \times_X X_j} (this is the analog of the intersections {U_i \cap U_j}), then we can reconstruct a sheaf on {X}. When the {X_i \rightarrow X} are open immersions, then this is just the previous result. When they are only required to be flat and have images covering {X}, 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 {\mathfrak{C}} 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 {X \in \mathfrak{C}} and a collection of arrows {\{X_i \rightarrow X\}} is considered a cover. This notion must satisfy the following axioms:

  1. Any isomorphism {\left\{X \rightarrow Y\right\}} is a cover.
  2. If {\{X_i \rightarrow X\}} is a cover and {Y \rightarrow X} is a morphism, then {X_i \times_X Y \rightarrow Y} is a cover. Covers are stable under base-extension.
  3. If {\left\{X^i_{j} \rightarrow X_i \right\}} is a cover for each {i} and {\left\{X_i \rightarrow X\right\}} is a cover, then {\left\{X^i_{j} \rightarrow X, \forall i,j \right\}} is a cover of {X}.

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 {\left\{X_i \rightarrow X\right\}} is a cover if and only if each {X_i \rightarrow X} is an open immersion (i.e. isomorphism onto an open subset) and whose images cover {X}. The axioms say, in English:

  1. The trivial cover is a cover, as we would expect; more generally, anything equivalent in this category to the trivial cover is a cover.
  2. Given a cover of {X}, its inverse image under {Y \rightarrow X} is a cover. Pull-backs are categorical version of inverse images.
  3. Covers are transitive.

As an example, here is an illustration of the axioms:

Proposition 3 Suppose {\left\{X_i \rightarrow X\right\}} is a family of morphisms which covers {X}. Suppose {Y_i \rightarrow X_i} is a family of isomorphisms. Then the family of {Y_i \rightarrow X_i \rightarrow X} covers {X}.

Proof: Indeed, this follows because each {Y_i \rightarrow X_i} is a cover of {X_i}, and covers are transitive. \Box

We can modify the previous example of a Grothendieck topology:

Example 1 Let {X} be a fixed topological space and {\mathfrak{C}} the category of open subsets of {X} together with the inclusion maps. This has fibered products (which correspond simply to intersections). That is, {U \times_X V = U \cap V}, as is easily seen. Then we can define a Grothendieck topology on {\mathfrak{C}} by saying that a bunch of arrows {\left\{U_i \rightarrow U\right\}} cover {U} if and only if the images cover {U}. 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 {S}, and let {\mathfrak{C} } be the category of schemes over {S}.

  • We can define a topology on {\mathfrak{C}} by saying that a bunch of {S}-morphisms of schemes {X_i \rightarrow X} “cover” {X} if and only if the union of the images of {X_i} is {X}. It is very easy to see that the first and the third conditions of a topology are satisfied. For the second, we need a little more. Suppose given a family of morphisms {X_i \rightarrow X} which is jointly surjective in the above sense, and {Y \rightarrow X} is any morphism. Then we have a family of pull-backs {Y_i = X_i \times_X Y \rightarrow Y}. By a basic fact in algebraic geometry, we have

    \displaystyle  \mathrm{im}(Y_i \rightarrow Y) = (X \rightarrow Y)^{-1}\mathrm{im} (X_i \rightarrow X),

    where we have used a funny notation for the pre-image. Since the images of the {X_i} cover {X}, it follows that the {Y_i \rightarrow Y} 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 can modify the above topology to require that the morphisms {X_i \rightarrow X} are flat; note that this is indeed a topology. Now this is much more interesting, since flat morphisms are reasonably well-behaved, but in practice one wants some sort of finiteness condition as well.
  • We can say that a family of morphisms {X_i \rightarrow X} is a cover if and only if each {X_i \rightarrow X} is an open immersion and the images cover {X}. Then this is really just the family Zariski topology ported to become a Grothendieck topology.

  • 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.