One of the really nice pictures in homotopy theory is the “chromatic” one, relating the structure of the stable homotopy category to the geometry of formal groups (or rather, the geometry of the moduli stack of formal groups). A while back, I did a series of posts trying to understand a little about the relationship between formal groups and complex cobordism; the main result I was able to get to was Quillen’s theorem on the formal group of $MU$. I didn’t understand too much of the picture then, but I spent the summer engaging with it and think I have a slightly better feel for it now. In this post, I’ll try to give a description of how a natural attack on the homotopy groups of a spectrum via descent leads very naturally to the moduli stack of formal groups and to the Adams-Novikov spectral sequence. (There are other approaches to Adams-type spectral sequences, for instance in these notes of Haynes Miller.)

1. Descent

Let’s start with some high-powered generalities that I don’t really understand, and then come back to earth. Consider an ${E_\infty}$-ring ${R}$; the most important examples will be ${R = H \mathbb{Z}/2}$ or ${R = MU}$. There is a map of ${E_\infty}$-rings ${S \rightarrow R}$, where ${S}$ is the sphere spectrum.

Let ${X}$ be a plain spectrum. Then, equivalently, ${X}$ is a module over ${S}$. Tensoring with ${R}$ gives an ${R}$-module spectrum ${ R \otimes X}$, where the smash product of spectra is written ${\otimes}$. In fact, we have an adjunction

$\displaystyle \mathrm{Mod}(S) \rightleftarrows \mathrm{Mod}(R)$

between ${R \otimes }$ and forgetting the ${R}$-module structure. As in ordinary algebra, we might try to apply the methods of flat descent to this adjunction. In other words, given a spectrum ${X}$, we might try to recover ${X}$ from the ${R}$-module ${R \otimes X}$ together with the “descent data” on ${X}$. The benefit is that while the homotopy groups ${\pi_* X}$ may be intractable, those of ${R \otimes X}$ are likely to be much easier to compute: they are the ${R}$-homology groups of ${X}$.

Let’s recall how this works in algebra. Given a faithfully flat morphism of rings ${A \rightarrow B}$ and an ${A}$-module ${M}$, then we can recover ${M}$ as the equalizer of

$\displaystyle M \otimes_A B \rightrightarrows M \otimes_A B \otimes_A B.$

How does one imitate this construction in homotopy? One then has a cosimplicial ${E_\infty}$-ring given by the cobar construction

$\displaystyle R \rightrightarrows R \otimes R \dots .$

The ${\mathrm{Tot}}$ (homotopy limit) of a cosimplicial object is the homotopyish version of the 1-categorical notion of an equalizer. In particular, we might expect that we can recover the spectrum ${X}$ as the homotopy limit of the cosimplicial diagram

$\displaystyle R \otimes X \rightrightarrows R \otimes R \otimes X \dots .$ (more…)

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…)

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.

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…)