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.

Prime-field restriction preserves actual comparison diagrams

PDF p. 4: Weight zero

Statement

For the forgetful functor $u : {S c h}_{k}^{q c q s} \to C_{F}$ , the completed weight-zero motivic functor is $u^{*} Z (0)_{m o t, F}$ . Precomposition transports natural maps, cartesian diagrams and equivalences of source $D (Z)$ -valued functors to actual natural maps, cartesian diagrams and equivalences on qcqs $k$ -schemes. It requires no assertion that cdh sheafification commutes with changing the base field.

Facts & Assumptions

Given: The hypotheses in the statement.

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

[F1]

Motivic complexes and cohomology defines the actual restriction; The source qcqs cdh site provides its source objects.

Proof

1.1F1construct

The completed motivic definition is explicitly restriction from the prime field. Precomposition evaluates a source diagram at $u (X)$ , without changing any value or arrow. Limits and equivalences in a functor category are computed objectwise, so cartesian diagrams and equivalences are retained.

2.1F1step 1.1∎

A morphism of $k$ -schemes is also an $F$ -morphism. The same source naturality square therefore remains commutative after restriction. These observations give the actual diagram and all its compatibility maps, without comparing the big $k$ - and big $F$ -topoi.

Depends on

Used by · 2

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.