In this post, I’d like to describe a toy analog of the Sullivan conjecture. Recall that the Sullivan conjecture considers (pointed) maps from into a finite complex, and states that the space of such is contractible if is finite. The stable version replaces with the Eilenberg-MacLane spectrum:

Theorem 13Let be the Eilenberg-MacLane spectrum. Then the mapping spectrum

is contractible. In particular, for any finite spectrum , the graded group of maps .

In the previous post, I sketched a proof (from Ravenel’s “Localization” paper) of this result based on a little chromatic technology. The spectrum is **dissonant**: that is, the Morava -theories don’t see it. However, any finite spectrum is **harmonic**: that is, local with respect to the wedge of Morava -theories. It follows formally that the spectrum of maps is contractible (and thus the same with replaced by any finite spectrum). The non-formal input was the fact that is in fact harmonic, which requires a little work.

In this post, I’d like to sketch an earlier proof of the above theorem. This proof is based on the Adams spectral sequence. In fact, the proof runs parallel to Miller’s proof of the Sullivan conjecture. There is a classical Adams spectral sequence for computing , with page given by

with the (mod ) Steenrod algebra.

It turns out, however, for purely algebraic reasons, that the term is trivial. Miller’s proof of the Sullivan conjecture relies on more complicated algebra to show that the unstable version of all this has the same vanishing property at . Most of this material is from Margolis’s *Spectra and the Steenrod algebra.*

**1. The Adams spectral sequence**

Let’s start by describing a method for calculating when is any connective spectrum with finitely generated homology. The Adams spectral sequence is based on a cosimplicial resolution of

whose totalization (under these hypotheses) is the -adic completion . In particular, we have

which leads to a spectral sequence whose -page can be identified as

The groups are computed in the category of modules over the Steenrod algebra .

Of course, we wanted to begin with, while we have instead of . The good news is that we have an arithmetic square

Observe that the spectrum of maps from into a rational or -complete (for ) spectrum is trivial. The spectrum of maps from into is thus seen to be the same as the spectrum of maps from into . So we can restate the Adams spectral sequence:

**2. Injectivity of the Steenrod algebra**

The surprising result that leads to this computation is the fact that the Steenrod algebra is injective as a module over itself. This has an analog in the Sullivan conjecture: the cohomology of is injective in the category of unstable modules over the Steenrod algebra. This implies that the Adams spectral sequence for has its term concentrated on the bottom role with , and we get a complete description of maps out of :

Theorem 14Let be a connective spectrum with finitely generated homology. Then

There is no nonzero map of -modules . Thus this also implies the above result on finite spectra.

In the unstable case, there is a purely algebraic way to describe the *cohomology* of the mapping space , for a space.

In order to prove the above theorem, we are reduced by the Adams spectral sequence to showing the following purely algebraic result:

Proposition 15 (Adams, Margolis)The Steenrod algebra is injective in the category of (graded) modules over .

**3. Injectivity for finite-dimensional Hopf algebras**

Let be a finite-dimensional, graded, cocommutative, connected Hopf algebra over the ground field . “Connectedness” is the condition that . Let be the category of graded -modules.

The main goal of this section is to prove:

Proposition 16is injective in the category .

To start with, note that (with the -linear tensor product) is a symmetric monoidal category with monoidal product , thanks to the Hopf algebra structure on . The unit object is the field concentrated in degree zero. Another interesting object is itself, of course. A useful property of is the “shearing” isomorphism

with the following meaning: the first object is the internal tensor product (i.e., acts via the codiagonal) and the second object is an external tensor product (i.e., acts only on the first factor ).

*Proof:* The shearing map is determined by specifying a map of -vector spaces, and it is the evident map . In particular, this defines the natural transformation. We observe that the functors preserve colimits, so it suffices to check the result for connective and finite-dimensional. Any connective, finite-dimensional module has a filtration whose quotients are shifts of itself, so it suffices to check that the map is an isomorphism when . Here, though, it is obvious.

The category has dual objects as well (at least for finite-dimensional objects). Given a finite-dimensional -module , one has a dual object where the -linear structure comes from the antipode on . To show that is injective, we can play with this duality and give a formal argument.

In fact, let be the full subcategory of finite-dimensional (hence dualizable) objects. The existence of duals shows that is anti-equivalent to itself. It suffices to show that is injective in to see that it is injective in by a straightforward argument.

Therefore, to show that is injective in , we can use duality and show that is projective in . Since, up to a grading shift, is a retract of , it follows that is a retract of , and it suffices to show that is projective. But is clearly projective. This completes the proof.

**4. Proof of injectivity**

Now let’s consider the case of the Steenrod algebra . Our goal is to show that is injective in the category of graded -modules. In other words, for any graded -module , we want to show that

for all and all . It suffices to assume that is finite-dimensional in each dimension. In fact, we may assume that is finitely presented as a module over .

The Steenrod algebra is not finite-dimensional, but it is a cocommutative Hopf algebra. Moreover, it is the union

where each is a finite-dimensional, graded, connected Hopf algebra and therefore self-injective by the previous section. Moreover, is free over by Milnor-Moore structure theory. It turns out that this is enough to prove that is self-injective with only a little extra work.

In order to do this, observe that there exists and an -module such that

so it suffices to show that

Again, this follows because is a direct sum of copies of (by Milnor-Moore structure theory), so that is injective over . (Because is finite-dimensional, the infinite direct sum of injectives is again injective.) This implies that the -groups vanish and proves the claim.

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