Chern character induces the rational isomorphism on AHSS E-two

We check exactness of rationalization explicitly. For an abelian group $M$, form fractions $(m,s)$ with positive integers $s$, identifying $(m,s)\sim (m^{\mathrm{\prime }},s^{\mathrm{\prime }})$ if $t(s^{\mathrm{\prime }}m-s{m}^{\mathrm{\prime }})=0$ for some positive integer $t$. Reflexivity and symmetry are immediate; for transitivity multiply the two witnessing relations by the other denominators and add, producing a positive integer witness. Addition by common denominator and multiplication by rational scalars respect this relation by the same cross multiplication, giving a rational vector space $S^{-1}M$. The map $m\otimes (a\mathrm{/}s)\mapsto (am,s)$ respects [F5]'s defining relations. Conversely $(m,s)\mapsto m\otimes (1\mathrm{/}s)$ is well defined: a witnessing relation makes the difference $(s^{\mathrm{\prime }}m-s{m}^{\mathrm{\prime }})\otimes 1\mathrm{/}(s{s}^{\mathrm{\prime }})$ equal $t(s^{\mathrm{\prime }}m-s{m}^{\mathrm{\prime }})\otimes 1\mathrm{/}(ts{s}^{\mathrm{\prime }})=0$. These maps are inverse on the generators, so $M\otimes \mathbb{Q}=S^{-1}M$. In particular $m\mathrm{/}s=0$ exactly when some positive integer kills $m$. If $L\to M\stackrel{f}{\to }P$ is exact and $f(m)\mathrm{/}s=0$, then $tf(m)=0$ for some $t>0$, so $tm$ has a preimage $l\in L$ and $m\mathrm{/}s$ is the image of $l\mathrm{/}(ts)$. Conversely the composite is zero. Injectivity and surjectivity are preserved by the same fraction criterion. Thus rationalization is exact.

The actual pair sequences of [F3] and [F2] give skeletal exact couples. Explicitly for either cohomology theory use $D_{a,b}=h^{-a-b-1}(X^{-a-1})$, $E_{a,b}=h^{-a-b}(X^{-a},X^{-a-1})$, with restriction, boundary and pair map as $i,j,k$, then reindex the output by $(p,q)=(-a,-b)$. Exactness is exactly the three corresponding portions of the pair sequences. For $HP$, direct sums over the even shifts are exact: every element has finite support, and preimages for a finite support can be chosen finitely. Its pair maps are the ordinary ones. The character on the skeletal groups commutes with all three arrows by [F2].

Tensor the K-theory exact couple of step 1.2 with $\mathbb{Q}$. Step 1.1 preserves each exactness identity, hence gives another exact couple. For every differential group, exactness applied to $0\to \ker d\to E\to \mathrm{im}d\to 0$ and $0\to \mathrm{im}d\to \ker d\to H(E,d)\to 0$ proves that kernels, images and homology commute with rationalization. Induction over derived couples therefore identifies its $r$th page with $E_r(K)\otimes \mathbb{Q}$, and its differential with $d_r\otimes 1$. The target couple consists of rational vector spaces, so the additive character extends uniquely by $x\otimes a\mapsto a\mathrm{ch}(x)$. It still commutes with $i,j,k$, and [F4] supplies the asserted page morphism.

Apply [F1] to the two actual theories and the character from [F2]. Each first-page map is the coefficient map on each cell. By [F3] the source coefficient is $\mathbb{Z}$ in even degree and zero in odd degree. By the dimension axiom the target coefficient has only its $H^0(\ast )$ summand in even degree and is zero in odd degree. The normalization in [F2] sends $1$ to $1$ in degree zero, and its fixed Bott transport gives the same assertion in every even degree, positive or negative. Thus the coefficient map after tensoring is $\mathbb{Q}\to \mathbb{Q}$, $a\mapsto a$, or $0\to 0$. There are finitely many cells in a column, so tensoring its finite product of coefficient groups is the same as taking their tensor products coordinatewise, using the finite projections and inclusions. Consequently the rationalized first-page map is an isomorphism in every bidegree.

The first-page map of step 2.2 is a cochain map by step 2.1. A bijective cochain map has a cochain inverse: conjugate the differential-commutation identity by its inverse. It therefore induces an isomorphism on homology, giving the asserted page-two isomorphism and its coefficient description. The same argument inductively also gives isomorphisms on all later pages. This argument needs no claim that an arbitrary theory's independently specified suspension and pair connector produce a normalized cellular differential.

By [F2] the absolute character preserves the skeletal kernels. Step 1.1 identifies the rationalized kernels and their quotients with the kernels and quotients for the rationalized theory. In [F6]'s stable formula a relative representative maps to its pair image on a skeleton, and a lift from X determines a unique filtration coset. The commuting pair and restriction squares of step 1.2 carry such a representative and lift to the corresponding ones for $HP$. Thus the stable page map is precisely the associated-graded map of the rationalized character.

If $X$ is empty or a column has no cells, [F1] gives zero first pages and hence zero subsequent pages. For a zero-dimensional complex only column zero remains. In every total degree there are at most $\dim X+1$ columns; no global bound on the coefficient rows is asserted. Odd rows have the zero isomorphism, not an omitted comparison, and every negative even degree is included by step 2.2. AC is exactly the inherited assumption [A1]; the fraction and finite-coordinate arguments introduce no further choice requirement. These checks complete the claim.