why $HF_p$(Eilenberg Mac Lane spectrum) smash X (connective spectrum with finite type cohomology groups) is a wedge of suspensions of $HF_p$?

Question

why $HF_p$(Eilenberg Mac Lane spectrum) smash $X$ (connective spectrum with finite type cohomology groups) is a wedge of suspensions of $HF_p$?

This is Prop 2.1.2 (g) in Ravenel's green book. Can anyone explain it a little?

Thanks1

Answer 1

Let me write $HK$ for the Eilenberg-Maclane spectrum, where $K = \mathbb{F}_{p}$ is the field with $p$ elements.

You know the homotopy groups of $X \wedge HK$, these are precisely the homology groups of $X$, which are dual to homology as you are over a field and cohomology of $X$ is finite-dimensional. Thus, you can choose a locally finite set of generators of these homotopy groups (as $K$-vector spaces) and use them to obtain a map

$\psi: Y = \bigvee _{i \in I} \Sigma ^{d_{i}} \mathbb{S} \rightarrow X \wedge HK$,

from a locally finite wedge of the sphere spectrum. You can now smash the above map with $HK$ and precompose with the structure map $id_{X} \wedge m: X \wedge HK \wedge HK \rightarrow X \wedge HK$, where $m: HK \wedge HK \rightarrow HK$ is the multiplication, to obtain

$\phi: Y \wedge K = \bigvee _{i \in I} \Sigma ^{d_{i}} HK \rightarrow X \wedge K$.

This is an isomorphism on homotopy groups by construction, as one can see as follows. We know the homotopy groups of $Y \wedge K$, as these are simply homology groups of $Y$ and are thus a free graded $K$-vector space on generators

$\Sigma ^{d_{i}} \mathbb{S} \simeq \Sigma ^{d_{i}} \mathbb{S} \wedge \mathbb{S} \rightarrow _{j_{i} \wedge u} Y \wedge HK$,

where $j_{i}: \Sigma ^{d_{i}} \mathbb{S} \rightarrow Y$ is the inclusion of the $i$-th summand and $u: \mathbb{S} \rightarrow HK$ is the unit map of the ring spectrum $HK$. This follows from the fact that the homology of the sphere spectrum is $K$-vector space generated by the unit $u: \mathbb{S} \rightarrow HK \simeq \mathbb{S} \wedge HK$ and the fact that homology takes wedge to direct sums.

Going back to the map $\phi: Y \wedge HK \rightarrow X \wedge HK$ that I claimed was a weak equivalence, we now know both spaces have the same homotopy groups and so it's enough to show that $\phi$ takes generators to generators. But the composition in question is

$\Sigma ^{d_{i}} \mathbb{S} \simeq \Sigma ^{d_{i}} \mathbb{S} \wedge \mathbb{S} \rightarrow _{j_{i} \wedge u} Y \wedge HK \rightarrow _{\psi \wedge id_{HK}} X \wedge HK \wedge HK \rightarrow _{id_{X} \wedge m} X \wedge HK$.

which is equal to

$\Sigma ^{d_{i}} \mathbb{S} \rightarrow Y \rightarrow _{\psi} X \wedge HK$,

as we know $HK \simeq HK \wedge \mathbb{S} \rightarrow _{id_{HK} \wedge u} HK \wedge HK \rightarrow _{m} HK$ is the identity, since $HK$ is a ring spectrum (ie. a monoid in the stable homotopy category). This is one of the generators we chose in the beginning and we are done.

I believe this argument works in a greater generality, ie. one only needs that the ring spectrum $R$ (here $HK$) in question is a field, ie. $\pi_{*}(R)$ is a graded field.