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.

Weight-zero sections and cohomology on the chosen general site

PDF p. 4: Weight zero

Statement

Assume the Axiom of Choice and fixed compatible site and universe size bounds for the constructions below. Composing $ρ_{0}$ with the source constant-sheaf identification gives a natural equivalence $Z (0)_{m o t, k} (X) ⟶ R Γ_{c d h}^{F} (X, Z) .$ It induces natural isomorphisms $H_{m o t}^{i, 0} (X, Z) ≅ H_{c d h, F}^{i} (X, Z)$ for every integer $i$ , and these groups vanish for $i < 0$ .

Facts & Assumptions

[A1]

Choice and the fixed compatible site and universe bounds are retained when applying Constant integer coefficients and exact derived evaluation to define the derived target in step 1.1 and obtain its negative-degree vanishing in step 2.1. The Axiom of Choice

Given: The hypotheses in the statement.

Goal: Establish the displayed construction or conclusion with its actual maps.

[F1]

Equivalence of the weight-zero pullback projection.

[F2]

Exact source weight-zero and constant-sheaf input.

[F3]

Constant integer coefficients and exact derived evaluation and Motivic complexes and cohomology.

Proof

1.1A1F1F2F3

Compose the actual equivalence of [F1] with the actual natural identification of $Z (0)_{c d h, F}$ in [F2]. Its target is derived evaluation of the ordinary constant sheaf on the site fixed in [F3], since that is explicitly the coefficient/site interpretation of the source input. This constructs the displayed map as a natural equivalence.

2.1A1F1F2F3step 1.1∎

Apply the cohomology functor. The left groups are those of the completed motivic definition, and the right groups are those defined in [F3]. Its negative-degree vanishing applies to the ordinary degree-zero constant sheaf. Naturality follows by applying the same functor to the natural equivalence.

Depends on

Used by · 1

Motivic weight zero on all qcqs schemes

Dependency tree · 3 levels

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