Third edge homomorphism

PDF p. 17: Corollary6.9 supported factor calculation

Assume the Axiom of Choice. Let $k$ be a field and let $X$ be a Noetherian $k$-scheme of Krull dimension $3$. Write $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^{3}K(X)$ for the third stage of the decreasing motivic filtration of the nonconnective $K$-theory spectrum of perfect complexes, and $K_0(X)={\pi }_{0}K(X)$. Its third image-filtration group is the subgroup $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^3{K}_0(X)=\mathrm{im}({\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{3}K(X)⟶K_0(X))\mathrm{.}$ The arrow is induced by the transition from stage three to stage zero, identified with $K(X)$. Let $p_3$ be the isomorphism from ${\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{3}K(X)$ to $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{6,3}(X,\mathbb{Z})=H^6(\mathbb{Z}(3{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(X))$ induced by the canonical map to the third graded cofiber and the specified graded comparison, as in Third-stage graded isomorphism. Here ${\pi }_0$ denotes the zeroth stable homotopy group, and $H^6$ is degree-six cohomology of the weight-three motivic complex.

The third edge homomorphism is the composite ${ϵ}_3:H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{6,3}(X,\mathbb{Z})\stackrel{p_3^{-1}}{\to }{\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{3}K(X)⟶F_{\mathrm{m}\mathrm{o}\mathrm{t}}^3{K}_0(X)\mathrm{.}$ The second arrow is the stage-to-$K_0$ homomorphism with its codomain restricted to its image. In particular, ${ϵ}_3$ is surjective.

1.1 The hypotheses in [F1] hold for the given $X$, so $p_3$ has a unique inverse homomorphism from $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{6,3}(X,\mathbb{Z})$ to ${\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{3}K(X)$. [F1, given]

2.1 By [F2], the second arrow in the displayed composite is a well-defined surjective homomorphism onto the stated image subgroup. Composing it with the inverse from step 1.1 defines ${ϵ}_3$. [F2, step 1.1]

3.1 For any element of the image subgroup, choose a preimage under the second arrow and apply $p_3$ to it. The resulting motivic cohomology class maps to the chosen element under ${ϵ}_3$. Thus ${ϵ}_3$ is surjective. No injectivity of the stage-to-$K_0$ map is needed for this conclusion. [F1, F2, step 2.1] ∎