Supported RR and the actual support-to-global application

The positive-coordinate supplier in [F1], under the assumptions in [F4], supplies a smooth affine pure three-dimensional neighbourhood $V$ of the closed point $x$. Its point immersion is regular of codimension three, so apply [F1] to $x\hookrightarrow V$ with $q=d=3$, $r=0$ and $a=n$. Using [F1] and excision gives $c_{3,x}(n{u}_x)=2n{\delta }_x$. Apply the generator equality in [F2] to obtain ${ϵ}_{3,x}{c}_{3,x}(n{u}_x)=2n{u}_x$. This proves additivity of this supported $c_3$ as well; no general additivity of higher Chern operations is assumed.

The first global square is the one constructed in [F2]. For the edge square, the support-to-global maps of $F^{3}K$, its graded cofiber, and $K$ commute with the actual projection and augmentation. Since both supported and global $p_3$ are isomorphisms, their inverses commute on ${\pi }_0$ as well. Compose with stage augmentation to get precisely the edge equality of the statement. In particular ${\iota }_K{u}_x=[i_{\ast }k(x)]$ lies in the third image filtration.

For the nonpure case, [F4] gives $X=X_3⨿Y$ with $x\in X_3$. Put $U=X\setminus \{x\}$ and $U_3=X_3\setminus \{x\}$, so $U=U_3⨿Y$. All these schemes and opens are Noetherian, hence qcqs; $U_3$ need not be affine. Extension by zero from either clopen piece is exact on perfect complexes and identifies its perfect category with the supported category on that piece. The K-localization sequence in [F5] for $U_3\subseteq U$ is split by zero extension from $U_3$. Consequently actual restrictions give $K(U)\simeq K(U_3)\times K(Y)$. The derived sheaf and clopen descent in [F5] give the corresponding product of motivic complexes in each nonnegative weight.

Starting at stage zero, the K-product of step 1.3 and the natural triangles of [F5] inductively give $F^{j}K(U)\simeq F^{j}K(U_3)\times F^{j}K(Y)$ for every $j\ge 0$: the stage-$j$ and layer-$j$ product equivalences imply the stage-$(j+1)$ equivalence by taking fibers. These are restriction maps, compatible with transitions, graded comparisons and augmentations. Together with the products on $X$ in [F4], they identify restriction from $X$ to $U$ with the product of restriction from $X_3$ to $U_3$ and the identity on $Y$. Taking fibers therefore identifies the supported augmented towers at $x$ on $X$ and $X_3$; applying the same argument to the motivic-complex restrictions identifies their supported complexes. This supplies the claimed support identifications on the possibly nonaffine open.

These restrictions preserve the residue objects and fundamental point classes. Apply steps 1.1–1.2 on the pure normal affine threefold $X_3$. The supported Chern operations and their support-to-global squares transport along step 2.1 by [F2]. Globally, the Chern operations are pointed and natural for restriction, so a zero-extended point class has zero Chern coordinate on $Y$. For the edge, the global $p_3$ maps on $X$ and $X_3$ are isomorphisms by [F3]. Given a motivic class with coordinates $(b_3,0)$, the stage class with coordinates $(p_{3,X_3}^{-1}{b}_3,0)$ maps to it by naturality of the stage projections and the product decompositions. Uniqueness of the inverse of $p_{3,X}$ identifies this as its inverse image; no isomorphism assertion on $Y$ is needed. Augmentation also preserves these coordinates. Thus both global squares and the supported identities extend from $X_3$ to $X$ exactly as stated.

Apply the two squares to $z=n{u}_x$ and its supported Chern image. In the pure case steps 1.1–1.2, and in the nonpure case step 3.1, give ${ϵ}_3{c}_3(n[i_{\ast }k(x)])={\iota }_K{ϵ}_{3,x}{c}_{3,x}(n{u}_x)=2n[i_{\ast }k(x)]$. This applies a support-forgetting map to a supported equality and never cancels one, so it remains valid when the global point class vanishes.