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Supported degree-seven vanishing
PDF p. 13: First paragraph of Proposition 6.1 proof
Statement
Assume the Axiom of Choice and the completed size conventions. Let be an affine finite-type threefold over an arbitrary algebraically closed field , let , and let be a closed integral subset contained in (in particular the curve of Proposition6.1). Then The quotient notation denotes the actual support fiber of the completed definition. The proof identifies the actual smooth restriction with Chow restriction before using its surjectivity.
Facts & Assumptions
The assumed The Axiom of Choice and fixed size bounds discharge the inherited hypotheses of the five supplied comparison and support prerequisites [F1] at their applications in steps 1.1–2.1. The exact-sequence deduction makes no additional choice.
Given: The hypotheses in the statement, with the Axiom of Choice and fixed universe bounds.
Goal: Construct the stated actual maps and prove their consequence.
The five fully authored local prerequisites in deps, under their exact AC, field, geometry and map conventions.
Proof
The geometry and actual restriction-diagram prerequisites apply over this perfect field, without any finite-generation assumption on k over the prime field. The support-fiber prerequisite supplies excision with the exact indexing and maps.
The fourth prerequisite proves surjectivity of the actual motivic degree-six restriction using cycle localization and the commuting diagram. The fifth applies the support long exact sequence and smooth degree-seven vanishing, then transports the zero group by excision. This is the asserted full statement.
Depends on
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.
Sources
- Fasel–Yakerson · p. 13, Proposition 6.1 proof