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Actual Chow localization gives surjectivity of the motivic arrow

PDF p. 13: Proposition 6.1 support argument

Statement

Assume the Axiom of Choice and the fixed universe and site-size bounds of the supplied comparisons. Let $k$ be an algebraically closed field, let $X$ be an affine finite-type threefold over $k$ , put $U = X_{s m o o t h / k}$ , let $T \subset X$ be a closed subset contained in $U$ , and put $V = U ∖ T$ . Use the reduced structure on $T$ when a scheme structure is needed. For the actual open restriction $V ↪ U$ , the map ${C H}^{3} (U) \to {C H}^{3} (V)$ is surjective. Under the preceding natural arrow comparison this is exactly the map $H_{m o t}^{6, 3} (U, Z) \to H_{m o t}^{6, 3} (V, Z)$ , which is therefore surjective.

Facts & Assumptions

Given: $k, X, U, T, V$ and the choice and size assumptions of the statement.

Goal: Prove surjectivity of Chow restriction and transfer it to this particular motivic restriction arrow.

[F1]

The supported-cycle fiber sequence and the identity-on-cycles identification of degree-zero homology with ordinary Chow groups, compatible with open restriction. Cycle complexes with fixed support, Degree zero and negative homology of the cycle complex

[F2]

The natural comparison for the actual restriction arrow under the intended geometry. A coherent comparison of the particular open restriction

[F3]

For the stated $k, X, U, T, V$ , the schemes $U, V$ are smooth separated finite type over the perfect field $k$ . The actual smooth open pair and its Nisnevich square

[F4]

The Axiom of Choice is retained as an assumption of the supplied comparison framework; the following homology and square arguments make no additional choice. The Axiom of Choice

Proof

1.1givenF1F3

By [F3], $U, V$ satisfy the smooth finite-type hypotheses of [F1]. Since $T$ is closed in $X$ and contained in $U$ , it is closed in $U$ and has open complement $V$ . The fixed-support cycle supplier gives the homological fiber sequence $z_{T}^{3} (U, ∙) \to z^{3} (U, ∙) \to z^{3} (V, ∙)$ . Its long exact homology sequence ends in $H_{0} z^{3} (U) \to H_{0} z^{3} (V) \to H_{- 1} z_{T}^{3} (U)$ . The last group is zero because the supported complex is concentrated in nonnegative homological degrees. Thus the first arrow is surjective. The identity-on-cycles identification of $H_{0}$ with ordinary Chow groups gives exactly the stated Chow restriction.

2.1F1F2F3F4step 1.1∎

Apply [F2] to this pair, retaining the choice and size assumptions recorded in [F4]. The comparison was constructed as an equivalence of actual restriction diagrams before taking $H^{6}$ . The square of its degree-six maps commutes and its vertical maps are isomorphisms. Surjectivity therefore transfers across that square to the actual motivic arrow, not an arbitrarily identified homomorphism.

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