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.

The actual finite-quotient map to cdh-local motivic cohomology

PDF p. 5: Prime-to-characteristic comparison

Statement

Let $F = Q$ or $F = F_{p}$ for a prime $p$ , and work on the category of all qcqs $F$ -schemes. For $j \geq 0$ , write $M_{j} = Z (j)_{m o t, F}$ for the supplied Elmanto–Morrow complex-valued presheaf, and $Z (j)_{c d h, F}$ for its specified classical-input cdh target. For every integer $n \geq 1$ invertible in $F$ , the objectwise derived quotient of the canonical motivic-to-cdh comparison is a natural equivalence $α_{j} : M_{j} / n \tilde{\to} Z (j)_{c d h, F} / n .$ The slash denotes the objectwise derived quotient $C \otimes^{L} Z / n = cofib (n : C \to C)$ , precisely as in the completed motivic definition. This is an equivalence of actual $D (Z)$ -valued presheaves on that same category and may be evaluated at every qcqs $F$ -scheme. If $k$ is any field with prime subfield $F$ , precomposition along the forgetful functor from qcqs $k$ -schemes retains this equivalence, with source the standing motivic complex and target the restriction of the specified prime-field cdh complex.

Facts & Assumptions

Given: The hypotheses in the statement.

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

[F1]

Exact cdh factorization inputs for the source étale map imports the actual canonical maps and their stated equivalences on the qcqs prime-field category.

[F2]

Elmanto–Morrow motivic complex input supplies the prime-field motivic functor and its fixed-field restriction. Motivic complexes and cohomology and Functorial finite coefficients for derived complexes fix the objectwise derived quotient.

Proof

1.1F1F2

If $F = Q$ , apply the canonical-map assertion of EM Theorem4.10(4), supplied by [F1], with $m = n$ . Its slash is derived quotient, the same construction as [F2].

1.2F1F2construct

If $F = F_{p}$ , [F1] supplies the equivalence $M_{j} [1 / p] \to Z (j)_{c d h, F} [1 / p]$ . Since $p$ is a unit in $Z / n$ , localization followed by derived tensor with $Z / n$ is just derived tensor with $Z / n$ : the equality follows from $Z [1 / p] \otimes_{Z}^{L} Z / n ≅ Z / n$ , with $Z [1 / p]$ flat. Tensoring that actual comparison thus gives exactly the stated canonical quotient map.

2.1F1F2step 1.1step 1.2∎

For each qcqs $F$ -scheme $X$ , steps 1.1–1.2 give an equivalence upon evaluation at $X$ . All constructions are objectwise and functorial, so these maps form the stated natural equivalence. Precomposing that same diagram with the forgetful functor for any field $k / F$ preserves its maps and equivalences; by [F2] its source is precisely the standing motivic quotient. The target remains the restriction of the prime-field cdh complex, without any unproved exchange of restriction and sheafification. This proves both scopes.

Depends on

Used by · 3

Dependency tree · 3 levels

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