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.

Low-degree consequences of a weight-one comparison

PDF p. 4: equation (2.3)

Statement

Assume the Axiom of Choice and the site-size bounds of Cohomology of shifted Nisnevich units. Let $X$ be a qcqs scheme over a field, and suppose a morphism $f : R Γ_{N i s} (X, G_{m}) [- 1] \to Z (1)_{m o t} (X)$ has cofiber supported in degrees $> 3$ . Then it induces $H_{N i s}^{i - 1} (X, G_{m}) ≅ H_{m o t}^{i, 1} (X, Z)$ for $i \leq 3$ . In particular the motivic groups are zero for $i \leq 0$ , global units for $i = 1$ , and the Picard group for $i = 2$ .

Facts & Assumptions

Given: $X$ , $f$ , the cofiber bound, Choice, and the site-size bounds.

Goal: Derive the displayed cohomology identifications.

[F1]

The cofiber bound gives an isomorphism on cohomology through degree three (Exact range of a cofiber connectivity bound).

[F2]

The source groups and their low degrees are identified in Cohomology of shifted Nisnevich units. Motivic cohomology is cohomology of the target complex (Motivic complexes and cohomology).

Proof

1.1F1F2given

Apply [F1] to $f$ . By [F2], the source of the resulting degree- $i$ isomorphism is $H_{N i s}^{i - 1} (X, G_{m})$ and the target is $H_{m o t}^{i, 1} (X, Z)$ .

2.1F2step 1.1∎

Substitute the computations of [F2] for $i \leq 2$ . This gives zero, units and the Picard group in the stated ranges, retaining the assumptions required by the supplier.

Depends on

Used by · 1

Motivic weight one

Dependency tree · 3 levels

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