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Rank and determinant on the second piece

PDF p. 11: Opening paragraphs of Lemma 5.3 proof

Statement

Assume the Axiom of Choice and the fixed site and universe bounds required by the low-weight and perfect K-theory constructions. Let $k$ be an algebraically closed field and let $X$ be a normal quasi-compact quasi-separated $k$ -scheme. For every $α \in F_{m o t}^{2} K_{0} (X)$ , the composites $F_{m o t}^{2} K_{0} (X) ⟶ K_{0} (X) \overset{p_{0}}{\to} H_{m o t}^{0, 0} (X, Z), F_{m o t}^{2} K_{0} (X) ⟶ F_{m o t}^{1} K_{0} (X) \overset{p_{1}}{\to} H_{m o t}^{2, 1} (X, Z)$ vanish. Consequently $α$ has rank zero and $c_{1}^{p e r f} (α) = 0$ . If in addition $α = [V] - [W]$ for vector bundles $V, W$ on $X$ , then $V$ and $W$ have equal rank at every point and their determinant line bundles are isomorphic. In particular these assertions hold on normal finite-type $k$ -schemes.

Here $K_{0} (X)$ is the zeroth homotopy group of nonconnective K-theory of perfect complexes, and $F_{m o t}^{j} K_{0} (X)$ is the image of the $j$ th motivic filtration stage in it. A perfect complex is locally quasi-isomorphic to a bounded complex of finite-rank vector bundles; its rank is the locally constant Euler characteristic of its derived fibers. A vector bundle is a finite locally free sheaf, and its determinant is its top exterior power on each constant-rank locus. Normal means that all local rings are integrally closed domains. Quasi-compact means every open cover has a finite subcover; quasi-separated means intersections of quasi-compact open subsets are quasi-compact. Finite type over $k$ means a finite affine cover by spectra of finitely generated $k$ -algebras.

The map $p_{0}$ is the actual zeroth stage-to-graded projection, using $F_{m o t}^{0} K (X) = K (X)$ . The first-stage augmentation identifies $π_{0} F_{m o t}^{1} K (X)$ injectively with $F_{m o t}^{1} K_{0} (X)$ on every qcqs field-scheme. The map $p_{1}$ here is the actual first stage-to-graded projection transported through this identification. Thus its construction applies throughout the stated qcqs scope and agrees with the existing finite-type projection convention. The first Chern class is the actual rank-reduced first-layer class, equivalently the perfect determinant followed by the actual weight-one Picard isomorphism. Choice means every family of nonempty sets has a choice function; the size bounds specify the sites and ambient universes of the cited constructions.

Facts & Assumptions

Given: The field $k$ , scheme $X$ , Choice, site and universe bounds, and $α \in F_{m o t}^{2} K_{0} (X)$ as stated.

Goal: Prove both projection composites vanish, then deduce rank zero and vanishing first Chern class, with the stated consequences for any bundle-difference representation.

[F1]

The decreasing motivic filtration has compatible stage maps to $K (X)$ , with stage zero identified with $K (X)$ , and nested image groups. For each stage, exactness makes the kernel of its stage-to-graded map on $π_{0}$ the image of the next stage. Motivic filtration and image groups Stage projections, kernels and image groups

[F2]

On every qcqs scheme over a field, the actual weight-zero map is the locally constant Euler rank on perfect $K_{0}$ . The first-stage augmentation is injective on $π_{0}$ and identifies its image with the rank-zero subgroup of $K_{0}$ . The actual first EM stage is the rank-zero K space on all qcqs schemes

[F3]

On the qcqs Nisnevich site over a field, the actual map from the rank-zero K-space to the weight-one Picard stack is the perfect-complex determinant map. On components it retains global determinant line classes, including on nonaffine schemes. The globally normalized first-layer Picard map is determinant

[F4]

Under Choice and the stated bounds, on every qcqs field-scheme one has $c_{1}^{p e r f} = γ_{X} \det_{X}$ , with $γ_{X} : Pic (X) \tilde{\to} H_{m o t}^{2, 1} (X, Z)$ the actual weight-one orientation. This class agrees with the actual bundle and virtual-bundle first Chern classes. In particular, on a bundle difference its determinant is the Picard class of $\det V \otimes (\det W)^{- 1}$ . The Picard group consists of isomorphism classes of line bundles under tensor product. First Chern class factors through determinant

[F5]

The assumed Choice and size bounds are used when invoking the filtration and perfect rank/Picard constructions; choosing a single lift of a given image element below does not itself require arbitrary-index choice. The Axiom of Choice

Proof technique: Transport only the first-stage map through the full-qcqs rank-fiber identification, apply consecutive stage exactness, and then use the actual determinant comparison.

Proof

1.1F1F2F3F5given

Invoke the filtration and rank/Picard constructions under [F5]. By [F2], the first-stage augmentation is an isomorphism onto its image group, so composing its inverse with the actual first stage-to-graded map defines $p_{1}$ on $F_{m o t}^{1} K_{0} (X)$ . This uses the injective first-stage augmentation on the stated qcqs scheme, without applying a finite-type comparison outside its hypotheses. Whenever the finite-type projection convention applies, both definitions use the inverse of the same augmentation and the same stage map, so they agree.

2.1F1F2step 1.1given

Choose a lift $a \in π_{0} F_{m o t}^{2} K (X)$ of $α$ , which exists by the image-group definition. Let $b$ be its image in $π_{0} F_{m o t}^{1} K (X)$ and let $c$ be the image of $b$ in $π_{0} F_{m o t}^{0} K (X) = K_{0} (X)$ . Compatible augmentations give $c = α$ . The element $b$ is the unique first-stage representative of $α$ by [F2].

3.1F1step 1.1step 2.1

The first stage-to-graded map sends $b$ to zero because it comes from stage two. The zeroth stage-to-graded map sends $c$ to zero because it comes from stage one. These are exactly the adjacent-stage kernels in [F1]. By the transport in step 1.1 and the augmentation compatibility in step 2.1, this proves $p_{1} (α) = 0$ and $p_{0} (α) = 0$ , hence both displayed composites vanish.

4.1F2F3F4step 2.1step 3.1

By [F2], $p_{0} (α) = 0$ says that $α$ has pointwise Euler rank zero. By [F3] the value of the actual first-layer map on its unique first-stage representative $b$ is $γ_{X} (\det_{X} α)$ , which is $c_{1}^{p e r f} (α)$ by [F4]. Step 3.1 makes that value zero. Thus the rank and first-Chern conclusions hold for arbitrary perfect classes in the second image group.

5.1F4step 4.1given∎

If $α = [V] - [W]$ , additivity of Euler rank and step 4.1 give equal ranks of $V$ and $W$ at every point. By [F4], $γ_{X} ([\det V \otimes (\det W)^{- 1}]) = c_{1}^{p e r f} (α) = 0$ . Injectivity of $γ_{X}$ makes this Picard class zero, equivalently $\det V ≅ \det W$ . Normal finite-type schemes over $k$ are qcqs, so the original finite-type special case is included. This proves every assertion.

The article's original normal qcqs scope is retained, including nonseparated schemes. No global bundle-difference representation or resolution property is assumed. The final assertion applies only when a representation is supplied. The original article's blanket representation premise remains a source proof defect; the rank and determinant conclusions do not require it. This proof uses the actual full-qcqs rank and Picard interfaces instead of extending the finite-type smooth-locus comparison by assertion. Their underlying constructions are used as supplier premises, without new acceptance or audit claims.

Scope and reading limits

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Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 3, Notation and conventions; pp. 8-10, Lemmas 4.1 and 5.2; p. 11, opening of the proof of Lemma 5.3