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Third edge on a curve complement

PDF p. 14: Remark 6.3

Statement

Assume the Axiom of Choice and the size conventions used for motivic cohomology with support. Let $X$ be an affine finite-type scheme of Krull dimension three over an algebraically closed field $k$ , and let $T \subset X$ be a closed integral curve contained in the smooth locus of $X \to Spec k$ . Put $U = X ∖ T$ . Then the third edge homomorphism $ϵ_{3} : H_{m o t}^{6, 3} (U, Z) ⟶ F_{m o t}^{3} K_{0} (U)$ is an isomorphism, and the stage-to- $K$ homomorphism $π_{0} F_{m o t}^{3} K (U) ⟶ K_{0} (U)$ is injective. No normality assumption on $X$ is made.

An integral curve is a reduced irreducible scheme of dimension one; the smooth locus consists of the points where the structure morphism to $Spec k$ is smooth. The spectrum $F_{m o t}^{3} K (U)$ is the third stage of the decreasing motivic filtration of nonconnective $K$ -theory, and $π_{0}$ denotes its zeroth stable homotopy group. The group $K_{0} (U) = π_{0} K (U)$ is the zeroth homotopy group of the $K$ -theory spectrum. The image-filtration group $F_{m o t}^{3} K_{0} (U)$ is the image of the displayed stage-to- $K$ map. Finally, $H_{m o t}^{6, 3} (U, Z)$ is degree-six cohomology of the weight-three motivic complex on $U$ ; the third edge map is the inverse of the third-stage graded projection followed by the stage map with codomain restricted to its image.

Statement references

Torsion freeness after removing a smooth-locus integral curve

Rational third-edge comparison

Third edge homomorphism

Facts & Assumptions

Given: The field $k$ , scheme $X$ , curve $T$ , complement $U$ , Axiom of Choice, and size conventions in the statement.

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

[F1]

Under exactly these geometry, choice and size hypotheses, Torsion freeness after removing a smooth-locus integral curve gives that $H_{m o t}^{6, 3} (U, Z)$ is torsion-free: an element killed by a positive integer is zero. The size conventions are those of Motivic and filtered K theory with support. The assumed Axiom of Choice, The Axiom of Choice, discharges the Choice premise of this supplier and its support constructions, and the Choice premises of the rational comparison in [F2] and the third-stage projection in [F4].

[F2]

Rational third-edge comparison applies to this complement without normality: it establishes that $U$ is Noetherian of dimension three over $k$ and that the actual third edge becomes an isomorphism after tensoring with $Q$ .

[F3]

For a homomorphism of abelian groups with torsion-free domain, Detecting injectivity by rationalization gives injectivity when its rationalization is injective; if the homomorphism is also surjective, it is an isomorphism.

[F4]

Under Choice for a Noetherian three-dimensional scheme over a field, Third-stage graded isomorphism gives the actual projection $p_{3} : π_{0} F_{m o t}^{3} K (U) \overset{≅}{\to} H_{m o t}^{6, 3} (U, Z)$ . By Third edge homomorphism, the edge is $p_{3}^{- 1}$ followed by the stage map to its image, and is surjective.

Proof

1.1F1F2given

The given $X$ , $k$ and $T$ satisfy the hypotheses of [F1] and the complement application of [F2]. Hence the domain of $ϵ_{3}$ is torsion-free and $ϵ_{3} \otimes_{Z} Q$ is an isomorphism, in particular injective.

2.1F3F4step 1.1

The edge map is surjective by [F4]. Applying [F3] to it and the two facts in step 1.1 shows that $ϵ_{3}$ is an isomorphism.

3.1F4step 2.1∎

Let $u : π_{0} F_{m o t}^{3} K (U) \to K_{0} (U)$ be the stage map. By [F4], $p_{3}$ is an isomorphism and $u$ is the composite of $p_{3}$ , $ϵ_{3}$ , and the inclusion of the image subgroup $F_{m o t}^{3} K_{0} (U)$ into $K_{0} (U)$ . Each factor is injective, using step 2.1 for the middle factor. Therefore $u$ is injective, proving both assertions.

Depends on

Used by · 1

Lifting the corrected class to the curve-supported third stage

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 · PDF p. 14, Remark 6.3 and proof of Proposition 6.2; p. 13, Proposition 6.1

For the curve complement in Proposition 6.1, the third edge map is an isomorphism and the third-stage map to K_0 is injective.

Hypotheses: An affine threefold over an algebraically closed field and an integral curve in its smooth locus, with the article’s standing conventions.

Reading limits: The cited remark, proposition and proof paragraph were read in the supplied article text. The argument below uses the stated completed supplier contracts for torsion-freeness, rational comparison and the actual edge map.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 14, Remark 6.3 and proof of Proposition 6.2; p. 13, Proposition 6.1