Smooth points generate the third motivic piece

By [F1], $X$ is a finite disjoint union of clopen integral components. The top-dimensional smooth-point subgroup is contained in $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^3{K}_0(X)$. It remains to prove the reverse inclusion. Let $\alpha$ be an arbitrary element of that image piece.

Apply [F2] to $\alpha$. It supplies a closed subset $C\subset W=X_{\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{h}}$ and a class $p\in F_{\mathrm{s}\mathrm{m},\mathrm{t}\mathrm{o}\mathrm{p}}^3{K}_0(X)$ such that $\beta =\alpha -p$ lies in $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^3{K}_0(X)$ and restricts to zero in $K_0(X\setminus C)$. For every three-dimensional component $X_e$, $C_e=C\cap X_e$ is an integral closed curve in $X_e$ contained in $W_e=W\cap X_e$; on every lower-dimensional component $C$ is empty.

By [F3], a representative of $\beta$ in ${\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{3}K(X)$ lifts to an element of the actual support group ${\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{3}K(X\mathrm{/}(X\setminus C))$. Restrict this lift to $W$ using the support equivalence in [F3]. Its image in ${\pi }_{0}K(W\mathrm{/}(W\setminus C))$, followed by the support-to-global map and the compatible comparison back on $X$, represents $\beta$. This uses the maps of support fibers, rather than identifying unrelated groups merely by isomorphism.

The finite clopen decomposition in [F1] restricts to $W$ and to its complement of $C$. Taking fibers of the componentwise stage and $K$ maps in [F5] decomposes both supported groups into the corresponding finite products. A component with empty support contributes the fiber of an identity map and hence zero on ${\pi }_0$. For each remaining component, $W_e$ is smooth of pure dimension three and $C_e\subset W_e$ is an integral closed curve of codimension two. Applying [F4] to the component of the lifted element shows that its supported $K_0$ image is a finite integer combination of the residue-sheaf classes at closed points of $C_e$.

Each such point is closed and smooth in $X_e$, hence in $X$: closedness follows because a closed point of the finite-type $k$-scheme $W_e$ has finite residue field over $k$, and smoothness is local on the source. By [F5], excision for a neighbourhood of each point identifies its supported class on $W_e$ with its supported class on $X_e$, compatibly with support-to-global maps; [F1] identifies subsequent clopen zero extension with the point class on $X$. Thus the image on $X$ of the combination in step 4.1 is a top-dimensional smooth-point class. Compatibility in step 3.1 shows that this image is $\beta$. Consequently $\alpha =p+\beta$ is in $F_{\mathrm{s}\mathrm{m},\mathrm{t}\mathrm{o}\mathrm{p}}^3{K}_0(X)$.

Since $\alpha$ was arbitrary, step 5.1 and the inclusion in step 1.1 prove the general equality. If $X$ is pure three-dimensional, all its components are three-dimensional, so [F1] identifies the top-dimensional subgroup with $F_{\mathrm{s}\mathrm{m}}^3{K}_0(X)$. This proves the article assertion.