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.

Restriction of the base field for weight one

PDF p. 4: Property (2)

Statement

Let $k$ be a field with prime subfield $F$ , and let $X$ be a quasi-compact quasi-separated $k$ -scheme. Then $X$ is a quasi-compact quasi-separated $F$ -scheme, and the weight-one complex assigned to $X$ in Motivic complexes and cohomology is the value at this same scheme of the Elmanto–Morrow prime-field functor. The small Nisnevich site of $X$ and its sheaf of units depend on $X$ , not on which of these two structure maps is retained.

Facts & Assumptions

Given: $k$ , $F$ , and $X$ as in the statement.

Goal: Identify the scheme, motivic value, and small-site data used after restriction of scalars.

[F1]

The motivic complex over $k$ is defined by restricting the Elmanto–Morrow functor on qcqs $F$ -schemes (Motivic complexes and cohomology, Elmanto–Morrow motivic complex input).

Proof

1.1given

Composition of $X \to Spec k$ with $Spec k \to Spec F$ changes neither the underlying scheme nor its quasi-compactness or quasi-separatedness. Thus $X$ belongs to the prime-field domain.

2.1F1step 1.1∎

By [F1], evaluating the restricted functor on $X$ is evaluation of the original functor on that same underlying $F$ -scheme. The objects and morphisms of the small Nisnevich site are étale schemes over $X$ and maps over $X$ ; its covers are specified by their underlying étale maps and residue-field lifting property. These data, and the units of the structure sheaves on them, are unchanged by forgetting the $k$ -structure. Hence the asserted identifications hold.

Depends on

Used by · 1

Motivic weight one

Dependency tree · 3 levels

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