The purpose of this post is to introduce the basic category that enters into the unstable Adams spectral sequence and the proof of the Sullivan conjecture: the category of unstable modules over the Steenrod algebra. Throughout, I’ll focus on the (simpler) case of ${p=2}$.

1. Recap of the Steenrod algebra

Let ${X}$ be a space. Then the cohomology ${H^*(X; \mathbb{F}_2)}$ has a great deal of algebraic structure:

• It is a graded ${\mathbb{F}_2}$-vector space concentrated in nonnegative degrees.
• It has an algebra structure (respecting the grading).
• It comes with an action of Steenrod operations

$\displaystyle \mathrm{Sq}^i: H^*(X; \mathbb{F}_2 ) \rightarrow H^{*+i}(X; \mathbb{F}_2), \quad i \geq 0.$

The Steenrod squares, which are constructed from the failure of strict commutativity in the cochain algebra ${C^*(X; \mathbb{F}_2)}$, are themselves subject to a number of axioms:

• ${\mathrm{Sq}^0}$ acts as the identity.
• ${\mathrm{Sq}^i}$ is compatible with the suspension isomorphism between ${H_*(X; \mathbb{F}_2), \widetilde{H}_*(\Sigma X; \mathbb{F}_2)}$.
• One has the Adem relations: for ${i < 2j}$,

$\displaystyle \mathrm{Sq}^i \mathrm{Sq}^j = \sum_{0 \leq 2k \leq i} \binom{j-k-1}{i-2k} \mathrm{Sq}^{i+j-k}\mathrm{Sq}^k. \ \ \ \ \ (3)$

In other words, there is a (noncommutative) algebra of operations, which is the Steenrod algebra ${\mathcal{A}}$, such that the cohomology of any space ${X}$ is a module over ${\mathcal{A}}$. The Steenrod algebra can be defined as

$\displaystyle \mathcal{A} \stackrel{\mathrm{def}}{=} T( \mathrm{Sq}^0, \mathrm{Sq}^1, \dots )/ \left( \mathrm{Sq}^0 = 1, \ \text{Adem relations}\right) .$

This much actually makes sense for any spectrum ${X}$, although the cohomology of a spectrum ${X}$ is no longer a ring. The cohomology of ${X}$ can be described as (graded) homotopy classes of maps

$\displaystyle X \rightarrow H \mathbb{F}_2$

for ${H \mathbb{F}_2}$ the Eilenberg-MacLane spectrum. The cohomology operations in ${\mathcal{A}}$ come from self-maps of ${H \mathbb{F}_2}$, and indeed ${\mathcal{A}}$ is the ring (under composition) of self-maps ${H \mathbb{F}_2\rightarrow H \mathbb{F}_2}$. The cohomology of any spectrum thus takes values in the category of ${\mathcal{A}}$-modules.

When ${X}$ is a space, though, one has a bit more. For one thing, there is a ring structure on the cohomomology of ${X}$ (essentially because ${X}$ has a diagonal map ${X \rightarrow X \times X}$, whereas a spectrum does not in general).

1. One has the unstability condition

$\displaystyle \mathrm{Sq}^i x = 0 \quad \text{if } i > \mathrm{deg}x.$

2. One has the Cartan formula

$\displaystyle \mathrm{Sq}^k(xy) = \sum_{i+j=k} \mathrm{Sq}^i x \mathrm{Sq}^j y.$

3. One has

$\displaystyle \mathrm{Sq}^{\mathrm{deg} x} x = x^2.$

In this post, I’ll focus on the “linear” part of these conditions (the one that can apply to ${\mathcal{A}}$-modules). The second two “non-linear” conditions that describe algebras will be discussed later.

Definition 25 A (graded) ${\mathcal{A}}$-module is unstable if it satisfies the condition ${\mathrm{Sq}^i x =0 , i > \mathrm{deg}x}$. There is a category of unstable ${\mathcal{A}}$-modules, denoted ${\mathcal{U}}$.

The algebraic input that goes into the proof of the Sullivan conjecture is certain very non-trivial facts about the category ${\mathcal{U}}$. For example, we saw in a previous post that the self-injectivity of ${\mathcal{A}}$ could be used to compute homotopy classes of maps out of ${H \mathbb{F}_2}$ very easily. Our goal is to describe an analog in the category ${\mathcal{U}}$. In particular, we will want to prove:

Theorem 26 Let ${V}$ be a finite-dimensional ${\mathbb{F}_2}$-vector space. Then the cohomology ${H^*(BV; \mathbb{F}_2)}$ is an injective object in ${\mathcal{U}}$.

Before this, the goal of the next couple of posts is to understand the basics of the category ${\mathcal{U}}$.

2. Free unstable modules

The category ${\mathcal{U}}$ of unstable ${\mathcal{A}}$-modules is a Grothendieck abelian category. However, is not a category of modules over a ring: it does not have a compact projective generator. However, it does have a set of compact projective objects which together generate ${\mathcal{U}}$.

Definition 27 Let ${F(n)}$ denote the free unstable ${\mathcal{A}}$-module on a generator in degree ${n}$.

In order words, this means that for any ${M \in \mathcal{U}}$, one has a functorial isomorphism

$\displaystyle \hom_{\mathcal{U}}(F(n), M) \simeq M_n.$

The existence of ${F(n)}$ follows abstractly from the adjoint functor theorem, and the universal property shows that it is a compact projective object. It’s also easy to see that they generate the category ${\mathcal{U}}$: if ${M \in \mathcal{U}}$, we can produce a surjection

$\displaystyle \bigoplus_{n} \bigoplus_{m \in M_n} F(n) \twoheadrightarrow M$

hitting each element of ${M_n}$ for each ${n}$.

Using Kuhn’s version of the Gabriel-Popescu theorem, we find:

Proposition 28 The category ${\mathcal{U}}$ is equivalent to the category of representations of the opposite to the subcategory ${\mathcal{S}}$ of free objects ${\left\{F(n)\right\}_{n \in \mathbb{Z}_{\geq 0}}}$.

In fact, the theorem would in general give us that ${ \mathcal{U}}$ is a localization of the category of representations. However, in this case the objects are compact and projective. In the previous post, we produced a localization functor

$\displaystyle F: \mathrm{Rep}(\mathcal{S}^{op}) \rightarrow \mathcal{U}$

which sent the functor (“representation”) represented by ${F(n)}$ to ${F(n)}$. Let’s prove that ${F}$ is fully faithful, which will show that the “localization” is an equivalence. Indeed, consider the class of ${X, Y \in \mathrm{Rep}(\mathcal{S}^{op})}$ such that

$\displaystyle \hom(X, Y) \rightarrow \hom(FX , FY)$

is an isomorphism. By definition, it includes the pairs of representable functors. Assume now representable ${X}$ and let ${Y}$ vary. Since ${F}$ commutes with colimits and since ${\hom(X, \cdot)}$ and ${\hom_{\mathcal{U}}(FX, \cdot)}$ commute with colimits (by compact projectivity), it follows that ${\hom(X, Y) \rightarrow \hom(FX, FY)}$ is an isomorphism for ${X}$ representable and ${Y}$ arbitrary. Now fix ${Y}$ arbitrary and let ${X}$ vary.

In the next post, we’ll see that the free objects $F(n)$ can be described fairly concretely, and use that to derive properties of the category $\mathcal{U}$.