We continue in the quest towards descent theory. Today, we discuss the fpqc topology and prove the fundamental fact that representable functors are sheaves.

We now describe another topology on the category of schemes. First, we need the notion of an fpqc morphism.

Definition 1A morphism of schemes is calledfpqcif the following conditions are satisfied:

is faithfully flat (i.e., flat and surjective)is quasi-compact.

Indeed, “fpqc” is an abbreviation for “fidelement plat et quasi-compact.” It is possible to carry out faithfully flat descent with a weaker notion of fpqc morphism, for which I refer you to Vistoli’s part of FGA explained.

As with many interesting classes of morphisms of schemes, we have a standard list of properties.

Proposition 2

Fpqc morphisms are closed under base-change and composition.If are fpqc morphisms of -schemes, then is fpqc.

*Proof:* We shall omit the proof, since the properties of flatness, quasi-compactness, and surjectivity are all (as is well-known) preserved under base-change, composition, and products. This can be looked up in EGA 1 (except for flatness, for which you need to go to EGA 4 or Hartshorne III).

So we have the notion of fpqc morphism. Next, we use this to define a topology.

Definition 3Consider the category of -schemes, for a fixed base-scheme. Thefpqc topologyon is defined as follows: A collection of arrows is said to be a cover of if the map is an fpqc morphism.

This implies in particular that each is a flat morphism. We need now to check that this is indeed a topology.

- An isomorphism is obviously an fpqc morphism, so an isomorphism is indeed a cover.
- If is a fpqc cover and , then the morphism is equal to the base-change , hence is fpqc.
- Suppose is a cover for each and is a cover, I claim that is a cover. Indeed, we have thatfactors throughand we know that each morphism in the composition is flat (since the coproduct of flat morphisms is flat) and quasi-compact (since the coproduct of quasi-compact morphisms is quasi-compact). Similarly for surjectivity. It follows that is an fpqc cover.

So we have another topology on the category of schemes, which is very fine in that it is finer than many other topologies of interest (e.g. the fppf and etale topologies, which I will discuss at some other point). (more…)