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Torsion freeness after removing a smooth-locus integral curve

PDF p. 13: Proposition 6.1 and proof

Statement

Assume the Axiom of Choice and the size conventions of Motivic and filtered K theory with support. Let $X$ be an affine finite-type threefold over an algebraically closed field $k$ , and let $T \subset X$ be a closed integral curve contained in the smooth locus $X_{s m o o t h}$ . Then $H_{m o t}^{6, 3} (X ∖ T, Z)$ is torsion-free.

A threefold here is a finite-type $k$ -scheme of dimension three; no reducedness, irreducibility, normality, or smoothness of $X$ is assumed. An integral curve is a reduced irreducible scheme of dimension one. The smooth locus consists of the points at which $X \to Spec k$ is smooth. A group is torsion-free if $n a = 0$ for a positive integer $n$ implies $a = 0$ . The motivic cohomology group is the degree-six cohomology of the weight-three motivic complex, as defined in Motivic complexes and cohomology. The Axiom of Choice is used through the two supported-cohomology suppliers, including their size and comparison constructions.

Facts & Assumptions

Given: An affine finite-type threefold $X$ over an algebraically closed field $k$ , a closed integral curve $T \subset X_{s m o o t h}$ , the Axiom of Choice, and the size conventions of Motivic and filtered K theory with support.

Goal: Prove that $H_{m o t}^{6, 3} (X ∖ T, Z)$ is torsion-free.

[F1]

Motivic and filtered K theory with support defines the weight-three support spectrum as the fiber of restriction from $X$ to $X ∖ T$ , with supported cohomology $H_{m o t, T}^{a, 3} (X, Z) = π_{- a} M_{3} (X / (X ∖ T))$ . On the qcqs schemes in this statement, Motivic complexes and cohomology identifies the corresponding ordinary groups with cohomology of the evaluated motivic complexes. Homotopy exactness of a spectral fiber sequence supplies the long exact homotopy sequence of this actual fiber sequence.

[F2]

Torsion freeness of top threefold motivic cohomology says that $H_{m o t}^{6, 3} (X, Z)$ is torsion-free for an affine finite-type threefold over an algebraically closed field, without extra hypotheses on $X$ .

[F3]

Under the given field, geometry, choice and size hypotheses, Divisibility of curve-supported top cohomology says that the image of the actual support-to-global map in degree six is divisible, and Supported degree-seven vanishing says that the supported group in degree seven is zero. A group is divisible when multiplication by every positive integer is surjective.

[F4]

Quotient by a divisible subgroup says that in a short exact sequence $0 \to D \to G \to A \to 0$ of abelian groups, if $D$ is divisible and $G$ is torsion-free, then $A$ is torsion-free. Short exact means that the first map is injective, the second is surjective, and the image of the first equals the kernel of the second.

Proof

technique · Apply the support long exact sequence and the divisible-quotient lemma

1.1F1given

Put $V = X ∖ T$ . Since $X$ is noetherian, its open subset $V$ is quasi-compact; both are quasi-separated. The fiber sequence of [F1], followed by its long exact homotopy sequence with the given indexing, yields the exact segment $H_{m o t, T}^{6, 3} (X, Z) ⟶ H_{m o t}^{6, 3} (X, Z) \overset{ρ}{\to} H_{m o t}^{6, 3} (V, Z) ⟶ H_{m o t, T}^{7, 3} (X, Z) .$ The first map is support-to-global and $ρ$ is restriction.

2.1F3step 1.1given

The hypotheses of both suppliers in [F3] hold for the given $X$ , $k$ and $T$ , with the stated choice and size conventions. Thus the last group in step 1.1 is zero, and the image $D$ of the first map is divisible. Exactness makes $ρ$ surjective and identifies its kernel with $D$ .

3.1F2F4step 2.1∎

Write $G = H_{m o t}^{6, 3} (X, Z)$ and $A = H_{m o t}^{6, 3} (V, Z)$ . The inclusion of $D$ into $G$ and $ρ$ form a short exact sequence $0 \to D \to G \to A \to 0$ . By [F2], $G$ is torsion-free; by step 2.1, $D$ is divisible. Applying [F4] proves that $A$ is torsion-free, as required.

Depends on

Used by · 1

Third edge on a curve complement

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Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 13, Proposition 6.1 and proof