TheoremStatement: Literature-sourcedProof: Literature-sourced

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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.

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Separated raw-cycle Zariski descent input

PDF p. 4: Smooth raw cycle comparison; PDF p. 16: Supported generic localization

Statement

Let $Y$ be a smooth quasi-projective finite-type scheme over a field, and $r \geq 0$ . Regard $C (W) = z^{r} (W, ∙)$ in cohomological degrees $- n$ , and let $a C$ be its degreewise sheafification on the small Zariski site of $Y$ . The canonical unit $C (Y) ⟶ R Γ (Y_{Z a r}, a C)$ is a quasi-isomorphism. The comparison is natural for open restriction.

Source contract

Given: The hypotheses above and the unnormalized simplicial Bloch convention.

Goal: Supply the established separated case used in finite-affine-cover induction.

[F1] MVW Proposition19.12 and its two-localization-row proof, together with Theorem19.11, supply this actual unit equivalence; the standing convention on printed p.5 is separated. Only the indicated quasi-projective cases are used.

External input

Import [F1] at the stated scope and reading limits. The nonseparated extension is a separate local deduction. No proof of the Brown–Gersten theorem or the deep moving lemma is claimed here.

Depends on

Used by · 1

Actual raw simplicial cycles satisfy descent without separatedness

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

Mazza, Voevodsky and Weibel, Lecture Notes on Motivic Cohomology, December 2005 author PDF · Standing convention printed p.5; Definition19.10, Theorem19.11, Proposition19.12 and proof, printed pp.168–169 (PDF pp.179–180)

In the source separated finite-type scheme convention, the raw Bloch cycle complex has the Zariski Mayer–Vietoris property and its canonical map to derived sections of its degreewise sheafification is a quasi-isomorphism.

Hypotheses: Use only smooth quasi-projective finite-type schemes over a field, including affine schemes and quasi-compact open subschemes of an affine scheme. Use the actual simplicial cycle convention, not a replacement definition by hypercohomology.

Reading limits: Read the standing convention and the complete named definition, theorem statement, proposition/proof and proof of19.1 in the primary xprova PDF. The Brown–Gersten and moving foundations are imported. No claim that this source itself includes nonseparated schemes.

Sources

Mazza, Voevodsky and Weibel, Lecture Notes on Motivic Cohomology, December 2005 author PDF · Standing convention printed p.5; Definition19.10, Theorem19.11, Proposition19.12 and proof, printed pp.168–169 (PDF pp.179–180)