March 10, 2011

The site of G-sets and the associated category of sheaves

Posted by Akhil Mathew under algebraic geometry, category theory | Tags: , , , |
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In the past, I said a few words about Grothendieck topologies and fpqc descent. Well, strictly speaking, I didn’t get very far into the descent bit. I described a topology on the category of schemes (the fpqc topology) and showed that it was a subcanonical topology, that is, any representable presheaf was a sheaf in this topology.

This amounted to saying that if {X' \rightarrow X} was a fpqc morphism of schemes, then to hom out of {X} was the same thing as homming out of {X'} such that the two pull-backs to {X' \times_X X'} were the same. If I had gotten further, I would have shown that to give a quasi-coherent sheaf on {X} (among other things) is the same as giving “descent data” of a quasi-coherent sheaf on {X'} together with an isomorphism between the two pull-backs to {X' \times_X X'} satisfying the cocycle condition. Maybe I’ll do that later. But there is a more basic “toy” example that I now want to describe of a site (that is, category with a Grothendieck topology) and the associated category of sheaves on it.

1. {G}-sets

Our category {\mathcal{C}} is going to be the category of left {G}-sets for a fixed group {G}; morphisms will be equivariant morphisms of {G}-sets. We are now going to define a Grothendieck topology on this category. For this, we need to axiomatize the notion of “cover.” We can do this very simply: a collection of maps {\left\{U_i \rightarrow U\right\}} is called a cover if the images cover {U}. Now, fiber products of {G}-sets are calculated in the category of sets, or in other words the forgetful functor

\displaystyle G-\mathbf{set} \rightarrow \mathbf{Sets}

commutes with limits (as it has an adjoint, the functor {S \mapsto G \times S}). Thus, taking pull-backs preserve the notion of covering, and it is easy to see the other axioms are satisfied too: if we have a cover of each of the {U_i} (which cover {U}), then collecting them gives a cover of {U}. Similarly, an isomorphism is a cover. This is obvious from the definitions.

2. Representable presheaves

So we indeed do have a perfectly good site. Now, we want a characterization of all the sheaves of sets on it. To start with, let us show that any representable functor forms a sheaf; that is, the topology is subcanonical. (In fact, this topology is the canonical topology, in that it is the finest possible that makes representable functors into sheaves.) (more…)

 

September 9, 2010

The fpqc topology

Posted by Akhil Mathew under algebraic geometry, category theory | Tags: , , , |
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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…)

 

September 7, 2010

Sheaves in Grothendieck topologies

Posted by Akhil Mathew under algebraic geometry, category theory | Tags: , , |
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It is possible to define sheaves on a Grothendieck topology. Before doing so, let us recall the definition of a sheaf of sets on a topological space {X}.

Definition 1 A sheaf of sets {\mathcal{F}} assigns to each open set {U \subset X} a set {\mathcal{F}(U)} (called the set of sections over {U}) together with “restriction” maps {\mathrm{res}^U_V: \mathcal{F}(U) \rightarrow \mathcal{F}(V)} for inclusions {V \subset U} such that the following conditions are satisfied:

  • {\mathrm{res}^U_U = \mathrm{id}} and for a tower {W \subset V \subset U}, the composite {\mathrm{res}^V_W \circ \mathrm{res}^U_V } equals {\mathrm{res}^U_W}.
  • If {\left\{U_i\right\}} is a cover of {U \subset X}, then the map \displaystyle \mathcal{F}(U) \rightarrow \prod \mathcal{F}(U_i) is injective, and the image consists of those families {f_i \in \mathcal{F}(U_i)} such that the restrictions to the intersections are equal \displaystyle \mathrm{res}^{U_i}_{U_i \cap U_j} f_i = \mathrm{res}^{U_j}_{U_i \cap U_j}

    In particular, this says that if we have a family of elements {f_i \in \mathcal{F}(U_i)} that satisfy the above gluing condition, then there is a unique {f \in \mathcal{F}(U)} which restricts to each of them.

(more…)

 

September 6, 2010

Descent part I: Grothendieck topologies

Posted by Akhil Mathew under algebraic geometry, category theory | Tags: , |
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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}. (more…)