Factor two in the supported third Chern map

PDF p. 17: Corollary6.9 supported factor calculation

Assume the Axiom of Choice and the fixed site and universe bounds of the motivic and support constructions. Let $X$ be a normal affine finite-type $k$-scheme of pure dimension three over an algebraically closed field $k$ of arbitrary characteristic, and let $x$ be a smooth closed point, with closed immersion $i:\mathrm{Spec}k(x)\hookrightarrow X$. With the actual support operations constructed above, $K_{0,x}(X)\stackrel{c_{3,x}}{\to }{H}_{\mathrm{m}\mathrm{o}\mathrm{t},\mathrm{x}}^{6,3}(X,\mathbb{Z})\stackrel{{ϵ}_{3,x}}{\to }{K}_{0,x}(X)$ is multiplication by $2$. More precisely, $c_{3,x}([i_{\ast }k(x)])=2{\delta }_x$ and ${ϵ}_{3,x}({\delta }_x)=[i_{\ast }k(x)]$, where ${\delta }_x$ is the positive fundamental motivic point class. These maps commute with support-to-global maps into the current global Chern class and third edge. Consequently the global composite ${ϵ}_3{c}_3$ is multiplication by $2$ on each residue-sheaf point generator of the third image filtration. This conclusion also holds on a normal affine finite-type $k$-scheme of dimension three that has lower-dimensional components, for every smooth closed point on a three-dimensional component: its supported groups are those on the pure clopen union $X_3$ of the three-dimensional components, and its global point class is extended by zero.