Splitting of $H\mathbb{Z}$-module spectra

Question

It is classical result of Adams that every $H\mathbb{Z}$-module spectra splits as a wedge of Eilenberg-MacLane spectra. Let me briefly recall what he writes about the proof.

Let $M$ be an $H\mathbb{Z}$-module spectrum. Adams constructs a map $$\alpha:\bigvee_k\Sigma^k S(\pi_kM)\rightarrow M$$ by taking the wedge of the maps $\Sigma^kS(\pi_kM)\rightarrow M$ inducing an isomorphism on $\pi_k$, where $SA$ denote the Moore spectrum on the abelian group $A$.

The map $\alpha$ induces a map of $H\mathbb{Z}$ by taking $\tilde{\alpha} = \mu \circ (1\wedge \alpha)$.

Now, $\tilde{\alpha}$ is without doubt a map of $H\mathbb{Z}$-modules, but why is it a weak equivalence?

Answer 1

Perhaps it helps to first think about how you can construct a map $\alpha_k : \Sigma^k S(\pi_k M) \to M$ inducing an isomorphism on $\pi_k$. Choose a free resolution $$ 0 \to \bigoplus_{j \in J} \mathbb{Z} \to \bigoplus_{i \in I} \mathbb{Z} \to \pi_k(M) \to 0 $$ and realize it in $H_k$ by a homotopy cofiber sequence $$ \bigvee_{j \in J} S^k \to \bigvee_{i \in I} S^k \to \Sigma^k S(\pi_k M) . $$ Mapping to $M$ you obtain an exact sequence $$ \dots \to [\Sigma^k S(\pi_k M), M] \to Hom(\bigoplus_{i \in I} \mathbb{Z}, \pi_k(M)) \to Hom(\bigoplus_{j \in J} \mathbb{Z}, \pi_k(M)) \to \dots $$ In particular, $$ [\Sigma^k S(\pi_k M), M] \to Hom(\pi_k(M), \pi_k(M)) $$ is surjective. Choose $\alpha_k$ so that it maps to the identity. Then $\pi_k(\alpha_k) : \pi_k(\Sigma^k S(\pi_k M)) \to \pi_k(M)$ is an isomorphism. (You should check this last claim.)

Using the $H\mathbb{Z}$-module structure on $M$, you can factor $\alpha_k$ as the Hurewicz map $$ h : \Sigma^k S(\pi_k M) \to H\mathbb{Z} \wedge \Sigma^k S(\pi_k M) \simeq \Sigma^k H(\pi_k M) $$ followed by $$ \tilde \alpha_k : \Sigma^k H(\pi_k M) \to M . $$ By the Hurewicz theorem, $\pi_k(h)$ is an isomorphism. (One way to see this is to show that $H\mathbb{Z}$ can be built as a CW spectrum from $S$ by only adding $n$-cells for $n\ge2$, which does not change $\pi_0$.) Thus $\pi_k(\tilde \alpha_k)$ is an isomorphism. Taking the wedge sum of the maps $\tilde \alpha_k$ for all integers $k$ gives the weak equivalence $\bigvee_k \Sigma^k H(\pi_k M) \to M$.