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}.

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