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The actual determinant fiber is a natural retract of the rank-zero K-space

PDF p. 11: Proposition 5.5; pp. 11–12

Statement

Assume the Axiom of Choice and fixed compatible site and universe size bounds for the constructions below. On the Nisnevich site of qcqs schemes over a field, put $S = Ω^{\infty} {Fil}^{1} K$ , $B = Ω^{\infty} {gr}^{1} K ≃ B G_{m}$ and $T = Ω^{\infty} {Fil}^{2} K$ . The actual first projection $d : S \to B$ is determinant. Its actual pointed section $κ : B \to S$ sends a line to $[L] - 1$ . There is a natural equivalence of spaces over $B$ , $T \times B ≃ S$ , with map $(t, b) \mapsto i (t) + κ (b)$ , and consequently a pointed retraction $r : S \to T$ of $i : T \to S$ . On underlying K-classes, $r (s) = s - κ (d (s))$ . For every qcqs $Y$ , both $π_{0} T (Y) \to π_{0} S (Y)$ and $π_{0} S (Y) \to K_{0} (Y)$ are injective. The space $T$ is the actual homotopy fiber, retaining determinant-trivialization data.

Facts & Assumptions

[A1]

Choice and the fixed compatible site and universe bounds are retained for The globally normalized first-layer Picard map is determinant in [F1] and The actual first EM stage is the rank-zero K space on all qcqs schemes in [F2]. Step 1.1 uses their actual rank and determinant maps and homotopies; step 3.1 uses their evaluated fiber identifications. The translation and retraction arguments then use these supplied maps. The Axiom of Choice

[F1]

The globally normalized first-layer Picard map is determinant supplies the actual pointed maps $d, κ$ and the pointed homotopy $d κ ≃ {i d}_{B}$ , not merely equality on Picard groups. Motivic weight one identifies $B$ as the actual Picard 1-type with its additive structure.

[F2]

Motivic filtration and image groups makes $d$ a map of grouplike infinite-loop spaces and $T$ its homotopy fiber. The actual first EM stage is the rank-zero K space on all qcqs schemes identifies $S$ with the rank fiber over the discrete locally constant rank sheaf.

Proof

Given: These actual sheaf maps, their homotopies, AC and the existing universe bounds.

Goal: Construct the fiber retraction and prove injectivity without a fiber-preservation assertion for a plus map.

1.1A1F1F2

Define $Φ : T \times B \to S$ by addition of $i (t)$ and $κ (b)$ . Additivity of $d$ and the chosen homotopy in [F1] make this a map over $B$ . On the fiber over any generalized point $b$ , translation by $κ (b)$ is an equivalence from the fiber over zero to the fiber over $b$ ; its inverse is translation by $- κ (b)$ , using that same homotopy. Therefore $Φ$ is an equivalence over $B$ . This argument is internal to the sheaf infinity-topos and hence is natural on restriction. It does not require $κ$ to be additive.

2.1step 1.1

Take $r$ to be the first component of $Φ^{- 1}$ . Its underlying expression is $s - κ (d (s))$ with the nullhomotopy furnished by $d κ ≃ i d$ . Because $κ$ is pointed, the restriction over zero identifies $r i$ with identity, including the fiber path. This constructs an actual homotopy-fiber map rather than choosing a lift of an object class.

3.1A1F1F2step 1.1step 2.1∎

Evaluation of the section makes $π_{1} S (Y) \to π_{1} B (Y)$ surjective. The fiber exact sequence thus makes $π_{0} T (Y) \to π_{0} S (Y)$ injective; equivalently use the product equivalence. The rank target in [F2] is discrete on derived sections, so its $π_{1}$ vanishes and the rank-fiber sequence makes $π_{0} S (Y) \to K_{0} (Y)$ injective. No assertion about generation by global bundles is involved.

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