Rational Chern character isomorphism for finite CW complexes

By [F1] the character induces a morphism of the two Atiyah-Hirzebruch spectral sequences, compatible with the filtrations; by [F2] this morphism is an isomorphism on the second page, coefficientwise, in both parities.

By [F4] the two spectral sequences strongly converge to the filtered abutments and the filtrations are finite and exhaustive, because a finite CW complex has a bounded skeletal filtration; the target maps are the filtered maps induced by the character, compatible with the abutment identifications by [F1].

Applying the comparison theorem [F3] to the morphism of step 1.1, the associated graded maps are isomorphisms, and because both degree-$j$ filtrations are finite the filtered maps themselves are isomorphisms; hence $\mathrm{ch}\otimes \mathbb{Q}:K^j(X)\otimes \mathbb{Q}\to {⨁}_k{H}^{j+2k}(X;\mathbb{Q})$ is an isomorphism of filtered $\mathbb{Q}$-vector spaces.

The argument uses the isomorphism on $E_2$ and the compatible filtered target maps; it does not infer the filtered groups from a collapse, in accordance with [F5], which shows that an abstract collapsed page determines only the associated graded.

Multiplicativity. By [F1] the graded character is multiplicative on both parities, so the bijections of step 2.1 for $j$ even and $j$ odd are ring isomorphisms for the $\mathbb{Z}\mathrm{/}2$-graded products; naturality in $X$ is the naturality of the character.

Boundary cases. For $X$ a point the isomorphism reads $K^{2k}(\ast )\otimes \mathbb{Q}\cong \mathbb{Q}$ and $0$ in odd degrees, which is [F2]. For $j$ outside $[0,2\dim X]$ both sides are zero or reduced to finitely many terms, and the filtration endpoints are finite. The coefficient field $\mathbb{Q}$ is nonzero, so no zero-ring case arises; the empty complex has trivial $K$-theory and trivial cohomology. AC enters only through [A1].