Spectral rationalization and its homotopy groups

The small colimit in the statement exists by [F1] under [A1]. Smash commutes with that colimit, and smashing the finite diagonal and fold with $E$ preserves their biproduct description. Hence $E_{\mathbb{Q}}$ is naturally the colimit of $E\stackrel{2}{\to }E\stackrel{3}{\to }E\stackrel{4}{\to }\cdots$. Smash with a fixed spectrum preserves small colimits and is exact by [F1], proving the asserted functoriality and exactness. For $A={\pi }_{i}E$, [F2] now identifies ${\pi }_i(E_{\mathbb{Q}})$ with the group colimit of $A\stackrel{2}{\to }A\stackrel{3}{\to }A\stackrel{4}{\to }\cdots$, with the same transition maps by [F1].

By [F4], use the presentation $C=D\mathrm{/}\mathrm{im}(1-s)$ for the group colimit in step 1.1. Its relations $[{\iota }_r(a)]=[{\iota }_{r+1}(u_{r}a)]$ move every finite sum to a single common later stage, so every class has a finite-stage representative. If $a\in A_i$ and $b\in A_j$ have equal classes, write ${\iota }_i(a)-{\iota }_j(b)=(1-s)y$ with $y\in D$. Choose $N$ greater than $i$, $j$ and all nonzero coordinate indices of $y$. The homomorphism ${⨁}_{r=1}^N{A}_r\to A_N$ sending coordinate $r$ by $u_{rN}$ kills $(1-s)y$: for each coordinate of $y$, its two terms cancel because $u_{rN}=u_{r+1,N}{u}_r$. Thus $u_{iN}a=u_{jN}b$. Conversely, equality at a common stage gives equal classes by those relations. This proves eventual equality without injective transition maps. Number the copies of $A$ starting at $n=1$. Send $a$ in copy $n$ to $a\mathrm{/}n!$ in the fraction group of [F3]. These homomorphisms are compatible, since $(n+1)a\mathrm{/}(n+1)!=a\mathrm{/}n!$. They are jointly onto: for $a\mathrm{/}s$ take $n\ge s$, so $s$ divides $n!$, and use $(n!\mathrm{/}s)a$ in copy $n$. If $a\mathrm{/}n!=0$, then $ka=0$ for some positive $k$ by [F3]; choose $N\ge nk$ so $k$ divides $N!\mathrm{/}n!$. The image of $a$ in copy $N$ is zero, so its colimit class is zero. This proves the claimed isomorphism. It is natural because both formulas commute with homomorphisms, and at copy 1 the unit is $a\mapsto a\mathrm{/}1$. Exactness in [F3] also preserves kernels, images and quotients, by applying it to the associated short exact sequences. Positive multiplication on fractions has inverse $a\mathrm{/}s\mapsto a\mathrm{/}(ms)$, so [F1]'s equivalence detection makes $E_{\mathbb{Q}}$ rational. Finally, for rational $R$, mapping the colimit of step 1.1 into $R$ is the inverse limit of the spaces $\mathrm{Map}(E,R)$ with transition maps given by precomposition with positive multiplication. Naturality of multiplication identifies this with postcomposition by the corresponding equivalences of $R$. All transition maps are therefore equivalences. The inverse-sequence index in [F5] is small in the stipulated universes and has terminal object $1$, so its nerve is weakly contractible. The mapping-colimit universal property gives its coherent limiting cone. Applying the limit-cone direction of [F5] shows that its projection to the first term is an equivalence. The evaluation map is precisely restriction along $E\to E_{\mathbb{Q}}$. This proves the localization assertion and finishes the proof.