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.

An independent full equivalence into the truncated cdh localization

PDF p. 6: Proposition 2.6 proof

Statement

For every qcqs $k$ -scheme, every $j \geq 0$ and invertible positive integer $n$ , the actual source maps give a natural equivalence $γ_{X} : Z (j)_{m o t} (X) / n \overset{α_{X}}{\to} Z (j)_{c d h, F} (X) / n \overset{θ_{X}}{\to} (L_{c d h}^{F} τ^{\leq j} B_{j}) (X) .$ Here $F$ is the prime subfield; localization is on the completed entire qcqs $F$ -scheme site, and $B_{j} = R Γ_{e t} (-, μ_{n}^{\otimes j})$ . Both arrows are full equivalences before applying any cohomological truncation. This claim is independent of all pending Artin, singular-BL and étale-descent targets.

Facts & Assumptions

Given: The hypotheses in the statement.

Goal: Establish the stated exact map or descent conclusion.

[F1]

The actual finite-quotient map to cdh-local motivic cohomology. [F2] Exact cdh factorization inputs for the source étale map, The source qcqs cdh site and Motivic complexes and cohomology.

Proof

1.1F1F2

The completed finite-quotient supplier [F1] supplies the actual full equivalence $α$ on all qcqs prime-field schemes: Theorem4.10(4) in characteristic zero, and Theorem4.24(3) tensored with $Z / n$ in characteristic $r > 0$ with $r ∤ n$ . The completed factorization supplier supplies the full equivalence $θ$ by EM Theorem3.5(3).

2.1F1F2step 1.1∎

Their composite is a full equivalence and has the displayed actual source and target on evaluating at $X$ . Prime-field restriction is actual precomposition, so does not alter either arrow. All source maps and their domains are fixed by the completed suppliers; no assertion from a pending target or mere truncated comparison enters this argument.

Depends on

Used by · 3

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.