Zeroth projection is rank

PDF p. 10: Lemma 5.1

Assume the Axiom of Choice and the fixed compatible site and universe bounds of the supplied weight-zero and perfect K-theory constructions. Let $k$ be a field and $X$ a quasi-compact quasi-separated $k$-scheme. Under the canonical identifications $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{0,0}(X,\mathbb{Z})\cong H_{\mathrm{c}\mathrm{d}\mathrm{h},F}^0(X,\mathbb{Z})\cong \mathrm{LC}(\mathrm{\mid }X\mathrm{\mid },\mathbb{Z}),$ the actual zeroth filtration projection $p_0:K_0(X)\to H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{0,0}(X,\mathbb{Z})$ is the locally constant Euler rank. Here $F$ is the prime subfield of $k$, and the cdh group uses the supplied entire qcqs $F$-scheme site. The group $\mathrm{LC}(\mathrm{\mid }X\mathrm{\mid },\mathbb{Z})$ consists of functions on the underlying topological space that are constant on a neighbourhood of every point, with pointwise addition. No seminormality, normality, separatedness, finite-type or dimension hypothesis is needed for this rank identification.

In the original separated seminormal finite-type case over an algebraically closed field, the same identifications agree naturally with the constant-integer sections on the separated finite-type $k$-scheme cdh site and give $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{0,0}(X,\mathbb{Z})\cong H_{\mathrm{c}\mathrm{d}\mathrm{h}}^0(X,\mathbb{Z})\cong {\mathbb{Z}}^{{\pi }_0(X)}\mathrm{.}$ Here ${\pi }_0(X)$ is the finite set of connected components, and its components are open and closed. Thus the original componentwise statement, including disconnected and empty schemes, is retained. On general qcqs schemes the locally constant-function description is used; arbitrary functions on connected components are not asserted to be locally constant.

The spectrum $K(X)$ is the specified nonconnective K-theory of perfect complexes and $K_0(X)={\pi }_{0}K(X)$. A perfect complex is locally quasi-isomorphic to a bounded complex of finite locally free sheaves. Its Euler rank at a point is the alternating dimension of its derived fiber over the residue field. The projection is the actual stage-zero-to-graded map, with $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^{0}K(X)=K(X)$. A qcqs scheme is quasi-compact and quasi-separated. Seminormality in the finite-type specialization has the supplied local square-and-cube-root convention.

The actual first EM stage is the rank-zero K space on all qcqs schemes

Motivic weight zero on all qcqs schemes

Constant cdh sections on the two article sites

Integer cdh sections on seminormal schemes

The supplied article v1 text was read at its qcqs conventions on p. 3, Lemma 2.4 on p. 5 and Lemma 5.1 on p. 10. The current rank-zero-stage argument and weight-zero, stage-projection, site-comparison and integer-section contracts supply this deduction; their deep constructions are not independently reconstructed or accepted here. The original unrestricted smooth-locus proof premise remains a separate article defect. The four direct consumers were inspected: they use the rank identity on normal affine finite-type schemes and need no edits. Original coverage is unchanged.