Normal dimension strata and the actual residue class model

Finite type over a field makes $X$ Noetherian and gives finitely many irreducible components. Normality and [F1] make them disjoint and open as well as closed. Group them by dimension to obtain the finite clopen $X_e$; each is affine and all its components have dimension $e$.

A sheaf complex on a finite clopen union is exactly a tuple of complexes on its pieces. Perfection is local, so zero extension is exact, preserves perfect objects, restricts to identity on its own piece and to zero on the others. The supported category for a clopen piece is its perfect category. Apply [F2] to that piece and its clopen complement. The resulting fiber sequence is split by zero extension of the complement, and restrictions identify it with the split product sequence. Iteration gives the displayed equivalence of actual K-spectra and hence the direct-sum formula on ${\pi }_0$.

By [F3], the current point map sends $1$ to the globally defined perfect sheaf $i_{\ast }\kappa (x)$, and every local Koszul representative resolves that same object. On an affine scheme its bounded finite-projective representative maps, under the actual affine bridge, to its alternating Euler class. This is the classical $[\kappa (x)]$ convention in $K_0$ of projective modules. Zero extension of that sheaf agrees literally with the same residue sheaf on $X$ and has zero restriction to every other clopen piece, by step 2.1. These facts establish model and sign compatibility, without identifying coherent G-theory with perfect K-theory in a singular ambient scheme.