LemmaStatement: AI-generatedProof: 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.

The support long exact sequence kills degree seven

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 $k$ -scheme of dimension three, put $U = X_{s m o o t h / k}$ , and let $T \subset X$ be a closed subset contained in $U$ . Use the reduced structure on $T$ when a scheme structure is needed. Put $V = U ∖ T$ and $W = X ∖ T$ . For the actual motivic support fiber on $U$ in weight three, its degree-seven group is zero. Via excision the same holds for support $T$ on $X$ .

Facts & Assumptions

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

Goal: Construct the stated actual maps and prove their consequence.

[F1]

Excision and the exact fixed-support cycle fiber, Actual Chow localization gives surjectivity of the motivic arrow and Homotopy exactness of a spectral fiber sequence.

[F2]

Smooth motivic comparison, applied to the actual smooth $U$ .

[F3]

The assumed Axiom of Choice, The Axiom of Choice, discharges the foundational hypotheses of the support and Chow suppliers in [F1] and the smooth comparison in [F2], invoked in step 1.1. The following exact-sequence deduction makes no additional choice.

Proof

1.1F1F2F3

Apply the long exact cohomology sequence of the actual support fiber to obtain $H_{m o t}^{6, 3} (U) \to H_{m o t}^{6, 3} (V) \to H_{m o t, T}^{7, 3} (U) \to H_{m o t}^{7, 3} (U)$ . The first arrow is surjective by the preceding local Chow-restriction lemma. The last group is zero by the completed smooth comparison: $2 \cdot 3 - 7 = - 1$ and the cycle complex has no negative homology.

2.1F1F2step 1.1∎

Exactness gives zero for the middle supported group: its map to the zero last group has kernel equal to the image of the preceding zero connecting map. The support-fiber lemma gives the actual excision equivalence to support on $X$ , proving the claim there as well.

Depends on

Used by · 1

Supported degree-seven vanishing

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.