I am trying to understand Balmer's classification of the spectrum of the category $\mathsf{Sp}^\text{fin}$ of finite spectra.
The inclusion $\mathsf{Sp}^\text{fin}_{(p)} \subseteq \mathsf{Sp}^\text{fin}$ of finite $p$-local spectra into finite spectra is exact in the sense that it preserves finite products, cofiber sequences and suspensions. So the only thing what could possibly fail is being closed under retracts. But I did not manage to find a counterexample.
If I understand Balmer correctly it cannot be thick however, since the Moore-spectrum $S\Bbb S_{(p)}$ is $p$-local but not torsion, hence the thick subcategory generated by it is the whole of $\mathsf{Sp}^\text{fin}$.
It seems like I don't quite understand how to pass from the Balmer-spectrum to thick subcategories and back. For the moment my reasoning is as follows: If it were thick, it should be a radical tensor ideal, hence a closed subset of the Balmer-spectrum. I think the only good candidate is the closed subset given by the union over the $p$-strand $$\cal{P}_{p,\infty} \subseteq ... \subseteq \cal{P}_{p,n+1} \subseteq \cal{P}_{p,n} \subseteq ... \subseteq \cal{P}_{p,1}$$ But if this were the case, then every finit spectrum with vanishing $p$-localization had to be $p$-local and furthermore every finite $p$-local spectrum would have vanishing first Morava K-theory...
A spectrum is $p$-local iff its homotopy groups are $p$-local. Similarly, a spectrum $Y$ is finite iff it is bounded below and $H_n(Y;\mathbb{Z})$ is nontrivial for only finitely many $n$ and finitely generated in all degrees.
If $Y$ is $p$-local and $X$ a retract of $Y$, then $Y\simeq X\oplus \mathrm{cofib}(X\to Y)$ and it is easy to check that then the homotopy groups of $X$ are $p$-local. The same argument shows that a retract of a finite spectrum is finite.