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Weight-one K-one projection is determinant

PDF p. 9: Lemma 4.1 proof: exact row and restriction square

Statement

Assume the Axiom of Choice and fixed compatible site and universe size bounds for the big Nisnevich comparison and its small-site derived sections. Let $k$ be algebraically closed and let $X$ be a normal quasi-compact quasi-separated $k$ -scheme. For the specified motivic filtration on the nonconnective $K$ -theory of perfect complexes, the actual augmentation induces an isomorphism $π_{1} F_{m o t}^{1} K (X) \tilde{\to} K_{1} (X) .$ Under this isomorphism and the natural identification $H_{m o t}^{1, 1} (X, Z) ≅ Γ (X, O_{X})^{\times}$ , the actual first-stage projection $π_{1} F_{m o t}^{1} K (X) ⟶ H_{m o t}^{1, 1} (X, Z)$ is the determinant homomorphism $\det_{X} : K_{1} (X) \to Γ (X, O_{X})^{\times}$ . It is surjective, with section taking a unit $u$ to the $K_{1}$ -class of multiplication by $u$ on $O_{X}$ .

Here normal means that every local ring is an integrally closed domain; quasi-compact means that every open cover has a finite subcover, and quasi-separated means that intersections of affine opens are quasi-compact. A global unit is a global regular function with a multiplicative inverse. The spectrum $K (X)$ is the specified nonconnective $K$ -theory of perfect complexes, $K_{1} (X) = π_{1} K (X)$ , and the augmentation is the first transition $F_{m o t}^{1} K (X) \to F_{m o t}^{0} K (X) = K (X)$ . The first-stage projection is induced by the actual map from that stage to its weight-one graded piece, whose first homotopy group is $H_{m o t}^{1, 1} (X, Z)$ . This includes the article's normal finite-type schemes and its full normal qcqs scope; no finite-type, Noetherian or dimension hypothesis is used.

Facts & Assumptions

Given: An algebraically closed field $k$ , a normal qcqs $k$ -scheme $X$ , the Axiom of Choice, and the fixed site-size bounds in the Statement.

Goal: Prove the augmentation isomorphism, identify the actual first-stage projection with determinant, and exhibit its unit section.

[F1]

The filtration has a natural identification $F_{m o t}^{0} K (Y) = K (Y)$ for each qcqs $k$ -scheme $Y$ , and its transitions are natural in $Y$ (Motivic filtration and image groups). The stated qcqs hypothesis puts $X$ in this category (Article scheme conventions).

[F2]

For every such $Y$ the natural graded triangle at weight zero is $F^{1} K (Y) \to K (Y) \to Z (0)_{m o t} (Y)$ . Its homotopy long exact sequence identifies $π_{n} Z (0)_{m o t} (Y)$ with $H_{m o t}^{- n, 0} (Y, Z)$ (Motivic graded pieces and spectral sequence). The latter group vanishes in negative cohomological degrees (Motivic weight zero on all qcqs schemes).

[F3]

Under AC and the stated site-size bounds, the weight-one comparison naturally identifies $H_{m o t}^{1, 1} (Y, Z)$ with $Γ (Y, O_{Y})^{\times}$ for qcqs $Y$ ; the bounds fix small sites and universe sizes for derived sections (Motivic weight one, Derived global sections on a scheme site). The graded triangle at weight one gives the actual projection, naturally in $Y$ (Motivic graded pieces and spectral sequence).

[F4]

The globally normalized first-layer Picard map is determinant identifies the actual map $Ω^{\infty} F^{1} K \to Ω^{\infty} {gr}^{1} K ≃ B G_{m}$ with the perfect-complex determinant/Picard map on the big Nisnevich site of every qcqs field-scheme, including its actual loop map and descent data. Its Choice and compatible site/universe bounds are those assumed here. At $X$ , loops at the trivial line in $B G_{m} (X)$ are its global units. Thus taking $π_{1}$ uses the actual determinant rather than agreement of Picard groups on affine opens.

[F5]

Under AC, for every qcqs scheme $Y$ there is a natural determinant homomorphism $\det_{Y} : K_{1} (Y) \to Γ (Y, O_{Y})^{\times}$ . The natural map $s_{Y} (u) = [O_{Y} \overset{u}{\to} O_{Y}]$ satisfies $\det_{Y} \circ s_{Y} = id$ (Units split the K one determinant). AC is the assertion that every family of nonempty sets admits a choice function (The Axiom of Choice).

Proof

1.1F1F2given

Apply the weight-zero long exact sequence in [F2] to $X$ . The two groups surrounding $π_{1} F^{1} K (X) \to K_{1} (X)$ are $H_{m o t}^{- 2, 0} (X, Z)$ and $H_{m o t}^{- 1, 0} (X, Z)$ , both zero by [F2]. Hence the actual augmentation is an isomorphism, in the full stated qcqs scope.

2.1F3F4F5step 1.1

By [F4], the first-stage projection on spaces is the actual determinant/Picard map. Evaluate its pointed natural map at $X$ and take $π_{1}$ . Its target identifies with the global units by [F3] and the trivial-line automorphism description in [F4]. Under the augmentation isomorphism in step 1.1 the resulting map is precisely $\det_{X} : K_{1} (X) \to Γ (X, O_{X})^{\times}$ , with the actual automorphism normalization in [F5]. This identifies the specified projection, without choosing a smooth locus or a global bundle representative.

3.1F5step 1.1step 2.1∎

For every global unit $u$ , [F5] gives $\det_{X} ([O_{X} \overset{u}{\to} O_{X}]) = u$ . Combining this identity with step 2.1 proves surjectivity and supplies exactly the asserted unit section, using step 1.1 for the domain identification.

Maintenance reading limits

The supplied article extraction at Lemma 4.1 and its proof, PDF pp. 8–9, was read. The present extension uses the current qcqs global-Picard argument, the actual graded and low-weight interfaces, and the unit-section contract. Their underlying determinant and motivic inputs were not reconstructed or newly accepted. The direct low-homotopy consumer is repaired in the same proposal; no fresh external-source reading is claimed.

Depends on

Used by · 1

Low-stage homotopy equals its image

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 3, notation and conventions; pp. 8–9, proof of Lemma 4.1