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Canonical first motivic class on every perfect K zero class

PDF p. 8: Proposition3.2, perfect-complex first class and determinant

Statement

Assume the Axiom of Choice and the fixed compatible site-size and universe bounds of the specified perfect K-theory, derived-section and weight-one constructions. For every field $k$ and qcqs $X / k$ , the actual first EM layer defines a natural homomorphism $c_{1}^{p e r f} : K_{0} (X) \to H_{m o t}^{2, 1} (X, Z)$ as follows. Let $r (α)$ be the locally constant Euler rank, and let $s (r)$ be the virtual trivial-bundle class of that rank, formed on its finite clopen rank partition. Let $\tilde{α}$ be the unique element of $π_{0} {Fil}^{1} K (X)$ mapping to $α - s (r (α))$ . Set $c_{1}^{p e r f} (α) = π_{0} q_{1} (\tilde{α})$ , using the actual weight-one identification. Then $c_{1}^{p e r f} (α) = γ_{X} (\det_{X} α)$ for every perfect class. It agrees with the existing first class on actual and virtual vector-bundle classes under their canonical map to perfect K zero.

The size data consist of universes containing the schemes and perfect-category enhancements in use, the chosen small Nisnevich sites used for derived sections of units, and a suitably small skeleton of the qcqs schemes over the prime field in those universes for the big-site low-layer comparisons. All these categories and their sheaves are interpreted in a sufficiently large ambient universe. These are the supplied size conventions; the assertion retains every field and every qcqs scheme.

Statement references

The Axiom of Choice

Nonconnective K theory of perfect complexes

Derived global sections on a scheme site

The source qcqs cdh site

Motivic weight one

Facts & Assumptions

[A1]

Assume the Axiom of Choice as formulated in The Axiom of Choice. The compatible site and universe data are those of Derived global sections on a scheme site, The source qcqs cdh site and Nonconnective K theory of perfect complexes, with the weight-one site bounds of Motivic weight one. These premises are used through the supplied rank-zero and Picard comparisons in [F1], and through the perfect determinant and weight-one identification in [F2], at steps 1.1–3.1. The finite clopen rank-section construction itself uses only finite data.

[F1]

The actual first EM stage is the rank-zero K space on all qcqs schemes supplies the rank and unique lift. The globally normalized first-layer Picard map is determinant identifies the actual q₁ map with determinant on that rank-zero space.

[F2]

Determinant descends on the actual perfect K zero supplies the homomorphism $\det_{X}$ on all perfect classes. Motivic weight one supplies the natural isomorphism $γ_{X}$ . Agreement with the existing projective-relation and virtual bundle first classes matches actual/virtual bundle first classes.

Proof

Given: A field $k$ , a qcqs $k$ -scheme $X$ , Choice, the stated site-size and universe bounds, and $α \in K_{0} (X)$ in the supplied perfect model.

Goal: Justify the definition on all classes and prove the formula.

1.1F1F2constructA1

The rank function has finite image on qcqs $X$ . For each clopen rank stratum, extension of its trivial line by zero is a finite locally free sheaf (rank1 there, rank0 elsewhere). Integer combinations of these classes define $s (r)$ , including negative ranks. Refining partitions leaves the class unchanged by direct sums on clopen components. Thus $s$ is a natural additive section of rank. Its determinant is the trivial line: each component trivial bundle has trivial determinant there and in rank zero elsewhere, with the canonical gluing.

2.1F1step 1.1A1

The difference $α - s (r (α))$ has rank zero. By [F1] it has the unique stated first-stage lift, so the displayed definition uses no representative of $α$ and no global vector-bundle presentation. All constituent maps are natural and additive, hence so is $c_{1}^{p e r f}$ .

3.1F1F2step 1.1step 2.1A1∎

By [F1], the projected rank-zero lift equals $γ_{X} (\det_{X} (α - s (r (α))))$ . By the trivial determinant from step 1.1 and [F2], this is $γ_{X} (\det_{X} α)$ . On actual or virtual bundle classes, the latter expression equals their existing first coefficient by [F2]. Thus the comparison concerns the actual EM projection and original orientation, not a new map defined by the desired determinant equality alone.

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