I am currently reading Lurie's notes on chromatic homotopy theory and fail to see the following remark in lecture 26:
Remark 5. Let $X$ be a finite $p$-local spectrum. Then $H_\ast(X,\Bbb F_p) \simeq 0$ if and only if $X \simeq 0$. Moreover $H_k(X,\Bbb F_p)$ vanishes for almost all values of $k$.
The fact that $X$ is $p$-local amounts to saying that the canonical morphism $X \rightarrow S\Bbb Z_{(p)} \wedge X$ is an equivalence after smashing with the Moore-spectrum $S\Bbb Z_{(p)}$ (recall $\Bbb Z_{(p)} = \Bbb Z[q^{-1}\mid q\neq p]$). Meanwhile $H_\ast(X,\Bbb F_p) = \pi_\ast(X \wedge H\Bbb F_p)$. So the first claim becomes $X \simeq 0$ iff $X \wedge H\Bbb F_p \simeq 0$.
I don't know, whether finiteness of $X$ is crucial for this claim. If it is, I expect to use the fact, that $X$ is a cofiber of direct sums of suspensions of the $p$-local sphere $\Bbb S_{(p)} = S\Bbb Z_{(p)} \wedge \Bbb S \overset{???}= S\Bbb Z_{(p)}$. But in any case it looks like I should study $S\Bbb Z_{(p)} \wedge H\Bbb F_p$. I think by some universal coefficient type argument this has to be $S\Bbb F_p$, but then what? I also thought of using the fact that ordinary homology spectra split, but I wasn't able to conclude anything from it.
I feel like I lost track and am missing the tools to conclude a fact, which should be obvious. So I would greatly appreciate some guidance.
Let $X$ be a finite $p$-local spectrum. Suppose $X \not\simeq 0$. We shall show that $H_*(X; \mathbb{F}_p) \neq 0$. Without loss of generality, suppose $X$ is connective (otherwise, suspend it until it is). Since $X$ is not contractible, $\pi_*(X) \neq 0$. Since $X$ is $p$-local, $\pi_*(X) = \pi_*(X) \otimes \mathbb{Z}_{(p)}$. Thus $X$ has a homotopy group of minimal degree that is a nonzero $\mathbb{Z}_{(p)}$-module. Since $X$ is finite, this homotopy group is finitely generated (ruling out things like $\mathbb{Q}$). Thus, $\pi_*(X)$ contains a summand $A$ of the form $\mathbb{Z}_{(p)}$ or $\mathbb{Z}/p^m$.
By the Hurewicz theorem, this same summand appears in $H_*(X; \mathbb{Z})$. Now consider the cofiber sequence $H\mathbb{Z} \wedge X \xrightarrow{p} H\mathbb{Z} \wedge X \to H\mathbb{F}_p \wedge X$. In the long exact sequence of homotopy groups, the nonzero summand in integral homology identified above appears as $$\cdots \to ? \rightarrow A \xrightarrow{p} A \rightarrow ? \to \cdots.$$ But in either case $A = \mathbb{Z}_{(p)}$ or $\mathbb{Z}/p^m$, the multiplication by $p$ map has a kernel or a cokernel, which implies that $H_*(X; \mathbb{F}_p)$ is nonzero, as desired.