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Low-stage homotopy equals its image

PDF p. 8: Lemma 4.1; PDF p. 9: Conclusion of proof

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 an algebraically closed field and $X$ a normal quasi-compact quasi-separated $k$ -scheme. For every $0 \leq i \leq 2$ and $j \geq 0$ , the canonical homomorphism $π_{j} F_{m o t}^{i} K (X) ⟶ F_{m o t}^{i} K_{j} (X)$ is an isomorphism.

Normal means that every local ring of $X$ 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. Here $K (X)$ is the specified nonconnective $K$ -theory spectrum of perfect complexes, $K_{j} (X) = π_{j} K (X)$ , and $F_{m o t}^{i} K (X)$ is the $i$ th stage of its motivic filtration. The group on the right is the image of the stage-to- $K (X)$ map on $π_{j}$ , and the displayed homomorphism is its corestriction to that image. The fixed size bounds specify the small sites and ambient universes for derived sections; Choice says that every family of nonempty sets has a choice function.

Facts & Assumptions

Given: An algebraically closed field $k$ , a normal quasi-compact quasi-separated $k$ -scheme $X$ , Choice, the stated site-size bounds, and integers $0 \leq i \leq 2$ and $j \geq 0$ .

Goal: Prove that the canonical stage-to-image map is an isomorphism.

[F1]

The specified filtration has $F_{m o t}^{0} K (X) = K (X)$ , compatible transition maps, and image groups defined as in the statement. The given scheme is qcqs, so the stated suppliers apply (Article scheme conventions, Motivic filtration and image groups).

[F2]

For each $r \geq 0$ , the graded triangle is $F_{m o t}^{r + 1} K (X) \to F_{m o t}^{r} K (X) \to Z (r)_{m o t} (X) [2 r]$ . Its graded homotopy group in degree $n$ is $H_{m o t}^{2 r - n, r} (X, Z)$ , where motivic cohomology is the cohomology of the specified motivic complex. The triangle gives the corresponding long exact homotopy sequence (Motivic graded pieces and spectral sequence, Motivic complexes and cohomology, Stage projections, kernels and image groups).

[F3]

Weight-zero motivic cohomology vanishes in negative cohomological degrees. Under Choice and the stated site-size bounds, weight-one motivic cohomology vanishes in degrees at most zero and $H_{m o t}^{1, 1} (X, Z)$ is naturally the group of global units (Motivic weight zero on all qcqs schemes, Motivic weight one, Derived global sections on a scheme site).

[F4]

Under the given field, normality, qcqs, Choice and size hypotheses, the actual projection $π_{1} F_{m o t}^{1} K (X) \to H_{m o t}^{1, 1} (X, Z)$ is surjective: after the actual augmentation isomorphism and the global-units identification it is determinant, which has the unit section (Weight-one K-one projection is determinant). Choice is as stated in The Axiom of Choice.

Proof

technique · Use the exact sequences of the weight-zero and weight-one graded triangles and the surjectivity of the actual determinant projection

1.1F1given

Write $F^{r} = F_{m o t}^{r} K (X)$ and $q_{r} : π_{j} F^{r} \to K_{j} (X)$ for the map induced by the transition to stage zero. For $r = 0$ , [F1] identifies $q_{0}$ with the identity, so its corestriction to its image is an isomorphism. By definition, every $q_{r}$ is surjective onto its image; it remains to prove injectivity for $r = 1, 2$ .

2.1F2F3step 1.1

The weight-zero part of [F2] gives the exact segment $H_{m o t}^{- j - 1, 0} (X, Z) \to π_{j} F^{1} \overset{q_{1}}{\to} K_{j} (X)$ . Since $j \geq 0$ , the first group is zero by [F3]. Thus $q_{1}$ is injective, and its corestriction to $F_{m o t}^{1} K_{j} (X)$ is an isomorphism.

3.1F1F2F3step 2.1

The weight-one part of [F2] gives the exact segment $H_{m o t}^{1 - j, 1} (X, Z) \to π_{j} F^{2} \to π_{j} F^{1}$ . If $j \geq 1$ , its first group vanishes by [F3], so the transition $π_{j} F^{2} \to π_{j} F^{1}$ is injective. Its composite with $q_{1}$ is $q_{2}$ by compatibility of transitions; therefore $q_{2}$ is injective by step 2.1.

3.2F1F2F4step 2.1

If $j = 0$ , the weight-one exact sequence contains $π_{1} F^{1} \to H_{m o t}^{1, 1} (X, Z) \to π_{0} F^{2} \to π_{0} F^{1}$ . The first map is surjective by [F4], so exactness makes the connecting map zero and the following transition injective. Composing with the injective $q_{1}$ from step 2.1 shows that $q_{2}$ is injective also in degree zero.

4.1F1step 1.1step 2.1step 3.1step 3.2∎

Steps 1.1 through 3.2 establish injectivity of each relevant stage-to- $K$ map. Its corestriction is surjective onto the image by definition, hence is an isomorphism for all stated $i$ and $j$ .

Reading limits

The article's Lemma 4.1 and its proof on PDF pages 8–9 were read. The determinant identification and the motivic graded and low-weight results are used through the completed suppliers named above; their underlying constructions are not reconstructed here.

Qcqs maintenance

The original normal qcqs scope of Lemma 4.1 is retained for every stated nonnegative homotopy degree, using the current qcqs determinant-projection lemma in [F4]. No dimension, Noetherian or finite-type hypothesis is used in this argument. The sole direct consumer, second-Chern surjectivity in dimensions two and three, was inspected: its finite-type application and Choice/size premises satisfy the extended contract, so it requires no edit. No new external-source reading or independent acceptance is asserted.

Depends on

Used by · 1

Second Chern surjectivity in dimensions two and three

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 pp. 8–9, Lemma 4.1 and its proof