I'm studying the construction of Adams spectral sequence in Hatcher's "Algebraic Topology" chapter five; I'm in trouble with an "algebraic statement".
Let $X$ be a connective CW-spectrum of finite type, $\mathcal A$ be the Steenrod algebra and $H^*(X)$ the graded cohomology of $X$, regarded as an $\mathcal A$-module (the cohomology will be in $\mathbb Z_p$ with $p$ a fixed prime). Choose $\alpha_i$ generators of $H^*(X)$ as $\mathcal A$-module; each $\alpha_i$ can be represented by a map $X \rightarrow K(\mathbb Z_p, m)$, so these $\alpha_i$ determine a map $X \rightarrow K_0$, where $K_0$ is a wedge of Eilenberg-Maclane spectra, and $K_0$ has finite type (since $X$ has finite type too, so each $H^k(X)$ is finitely generated). The statement I'm in trouble with is the following: $H^*(K_0)$ is a free $\mathcal A$-module.
It has surely to be related with the fact that the cohomology of a $K(\mathbb Z_p, n)$ is a free $\mathcal A$-module in dimension $<2n$, but I feel like I'm getting losing myself among the definitions and I can't prove this assertion. Hatcher claims that this is related to the natural isomorphism $[X, \bigvee K(G, n_i)] \simeq \Pi_i [X, K(G, n_i)]$, but I can't figure this out. Thanks in advance.