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 1 A morphism of schemes {f: X \rightarrow Y} is called fpqc if the following conditions are satisfied:

  1. {f} is faithfully flat (i.e., flat and surjective)
  2. {f} 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

  1. Fpqc morphisms are closed under base-change and composition.
  2. If {f: X \rightarrow Y, g: X' \rightarrow Y'} are fpqc morphisms of {S}-schemes, then {f \times_S f': X \times_S X' \rightarrow Y \times_S Y'} 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). \Box

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

Definition 3 Consider the category {\mathfrak{C}} of {S}-schemes, for {S} a fixed base-scheme. The fpqc topology on {\mathfrak{C}} is defined as follows: A collection of arrows {\left\{U_i \rightarrow U\right\}} is said to be a cover of {U} if the map {\coprod U_i \rightarrow U} is an fpqc morphism.

This implies in particular that each {U_i \rightarrow U} is a flat morphism. We need now to check that this is indeed a topology.

  1. An isomorphism is obviously an fpqc morphism, so an isomorphism is indeed a cover.
  2. If {\left\{U_i \rightarrow U\right\}} is a fpqc cover and {V \rightarrow U}, then the morphism {\coprod( U_i \times_U V )\rightarrow V } is equal to the base-change {(\coprod U_i) \times_U V \rightarrow V}, hence is fpqc.
  3. Suppose {\left\{U^i_j \rightarrow U_i\right\}} is a cover for each {i} and {\left\{U_i \rightarrow U\right\}} is a cover, I claim that {\left\{U_j^i \rightarrow U\right\}} is a cover. Indeed, we have that\displaystyle  \coprod_{i,j} U^{j}_i \rightarrow U factors through\displaystyle  \coprod_{i,j} U^{i}_j \rightarrow \coprod_i {U_i} \rightarrow U and 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 {\left\{U^i_j \rightarrow U \right\}} 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…)