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 .
1. Recap of the Steenrod algebra
Let be a space. Then the cohomology has a great deal of algebraic structure:
- It is a graded -vector space concentrated in nonnegative degrees.
- It has an algebra structure (respecting the grading).
- It comes with an action of Steenrod operations
The Steenrod squares, which are constructed from the failure of strict commutativity in the cochain algebra , are themselves subject to a number of axioms:
- acts as the identity.
- is compatible with the suspension isomorphism between .
- One has the Adem relations: for ,
In other words, there is a (noncommutative) algebra of operations, which is the Steenrod algebra , such that the cohomology of any space is a module over . The Steenrod algebra can be defined as
This much actually makes sense for any spectrum , although the cohomology of a spectrum is no longer a ring. The cohomology of can be described as (graded) homotopy classes of maps
for the Eilenberg-MacLane spectrum. The cohomology operations in come from self-maps of , and indeed is the ring (under composition) of self-maps . The cohomology of any spectrum thus takes values in the category of -modules.
When is a space, though, one has a bit more. For one thing, there is a ring structure on the cohomomology of (essentially because has a diagonal map , whereas a spectrum does not in general).
- One has the unstability condition
- One has the Cartan formula
- One has
In this post, I’ll focus on the “linear” part of these conditions (the one that can apply to -modules). The second two “non-linear” conditions that describe algebras will be discussed later.
Definition 25 A (graded) -module is unstable if it satisfies the condition . There is a category of unstable -modules, denoted .
The algebraic input that goes into the proof of the Sullivan conjecture is certain very non-trivial facts about the category . For example, we saw in a previous post that the self-injectivity of could be used to compute homotopy classes of maps out of very easily. Our goal is to describe an analog in the category . In particular, we will want to prove:
Theorem 26 Let be a finite-dimensional -vector space. Then the cohomology is an injective object in .
Before this, the goal of the next couple of posts is to understand the basics of the category .
2. Free unstable modules
The category of unstable -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 .
Definition 27 Let denote the free unstable -module on a generator in degree .
In order words, this means that for any , one has a functorial isomorphism
The existence of 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 : if , we can produce a surjection
hitting each element of for each .
Using Kuhn’s version of the Gabriel-Popescu theorem, we find:
Proposition 28 The category is equivalent to the category of representations of the opposite to the subcategory of free objects .
In fact, the theorem would in general give us that 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
which sent the functor (“representation”) represented by to . Let’s prove that is fully faithful, which will show that the “localization” is an equivalence. Indeed, consider the class of such that
is an isomorphism. By definition, it includes the pairs of representable functors. Assume now representable and let vary. Since commutes with colimits and since and commute with colimits (by compact projectivity), it follows that is an isomorphism for representable and arbitrary. Now fix arbitrary and let vary.
In the next post, we’ll see that the free objects can be described fairly concretely, and use that to derive properties of the category .