PropositionStatement: Literature-sourcedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Third edge isomorphism

PDF p. 14: Proposition 6.2

Statement

Assume the Axiom of Choice. Let $k$ be an algebraically closed field and $X$ a normal affine threefold over $k$ , where threefold means a finite-type $k$ -scheme of Krull dimension three. Then the third edge homomorphism $ϵ_{3} : H_{m o t}^{6, 3} (X, Z) ⟶ F_{m o t}^{3} K_{0} (X)$ is an isomorphism. Moreover, the actual stage-to- $K$ homomorphism $π_{0} F_{m o t}^{3} K (X) \to K_{0} (X)$ is injective.

Here $K (X)$ is the nonconnective $K$ -theory spectrum of perfect complexes, $K_{0} (X) = π_{0} K (X)$ , and $F_{m o t}^{3} K (X)$ is the third stage of its decreasing motivic filtration. The group $F_{m o t}^{3} K_{0} (X)$ is the image of the stage-to- $K_{0}$ homomorphism. The group $H_{m o t}^{6, 3} (X, Z)$ is degree-six cohomology of the weight-three motivic complex. The map $ϵ_{3}$ is the inverse of the actual third-stage-to-graded isomorphism $p_{3}$ followed by the stage-to- $K_{0}$ homomorphism with codomain restricted to its image. A group is torsion-free if no nonzero element is killed by a nonzero integer.

Statement references

Third edge homomorphism

Third-stage graded isomorphism

Facts & Assumptions

Given: The Axiom of Choice, an algebraically closed field $k$ , and a normal affine finite-type $k$ -scheme $X$ of dimension three.

Goal: Prove that the specified third edge map is an isomorphism and that the actual third-stage map to $K_{0} (X)$ is injective.

[F1]

A finite-type scheme over a field is Noetherian. Under the Axiom of Choice, Third-stage graded isomorphism supplies the actual isomorphism $p_{3} : π_{0} F_{m o t}^{3} K (X) \to H_{m o t}^{6, 3} (X, Z)$ . By Third edge homomorphism, the stated edge map is $p_{3}^{- 1}$ followed by the stage map restricted to its image, and is surjective. The choice assumption used by these suppliers is as in The Axiom of Choice.

[F2]

For an affine finite-type threefold over an algebraically closed field, Torsion freeness of top threefold motivic cohomology states that $H_{m o t}^{6, 3} (X, Z)$ is torsion-free.

[F3]

Under the Axiom of Choice, Rational third-edge comparison states that the rationalization of the actual third edge map is an isomorphism for a Noetherian dimension-three scheme over a field.

[F4]

Detecting injectivity by rationalization states that a homomorphism with torsion-free domain is injective if its rationalization is injective.

Proof

technique · Detect the kernel of the edge map after rationalization, then use its relation to the actual third-stage map

1.1F1F2F3given

The given finite-type scheme $X$ is Noetherian and has dimension three. Thus the hypotheses of [F1], [F2], and [F3] hold. In particular, the domain of $ϵ_{3}$ is torsion-free, and $ϵ_{3} \otimes_{Z} Q$ is an isomorphism, hence injective.

2.1F1F4step 1.1

Apply [F4] to $ϵ_{3}$ . Step 1.1 shows that its domain is torsion-free and its rationalization is injective, so $ϵ_{3}$ is injective. It is surjective by [F1], and therefore is an isomorphism.

3.1F1step 2.1∎

Let $u : π_{0} F_{m o t}^{3} K (X) \to K_{0} (X)$ be the actual stage-to- $K_{0}$ homomorphism and let $v$ be the same map with codomain restricted to its image. By [F1], $v = ϵ_{3} \circ p_{3}$ . Both factors are isomorphisms by [F1] and step 2.1, so $v$ is injective. The inclusion of its image into $K_{0} (X)$ is injective; consequently $u$ is injective. This proves both assertions.

Depends on

Used by · 2

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.

External dependencies

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · Proposition 6.2 and its proof, PDF p. 14

For a normal affine threefold over an algebraically closed field, the third edge map is an isomorphism and the third-stage map into K_0 is injective.

Hypotheses: X is a normal affine threefold over an algebraically closed field, with the article’s motivic filtration and conventions.

Reading limits: The proposition and proof paragraph were read in the supplied article text. The proof below invokes the completed local supplier contracts for their substantive inputs.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · Proposition 6.2 and its proof, PDF p. 14