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 exact cdh-local-to-objectwise étale map

PDF p. 5: Prime-to-characteristic comparison

Statement

Let $F = Q$ or $F_{p}$ , let $j$ be a nonnegative integer and let $n \geq 1$ be an integer invertible in $F$ . Work on the qcqs prime-field cdh site of EM §3.3 p.14, with Nisnevich covers and abstract blowup families whose closed immersion and proper map are finitely presented. Let $L_{c d h}$ be the source derived sheafification, left adjoint to the inclusion of derived cdh sheaves. Since $B_{j} = R Γ_{e t} (-, μ_{n}^{\otimes j})$ is a cdh sheaf, the truncation map $ι : τ^{\leq j} B_{j} \to B_{j}$ has its canonical adjoint extension $ϵ_{j} : L_{c d h} τ^{\leq j} B_{j} ⟶ B_{j} .$ Its composite with the sheafification unit is $ι$ . This defines the actual source map; it asserts no objectwise range from a local equivalence alone.

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 supplies the exact derived cdh-localization/descent data used by the source proof.

Proof

1.1F1construct

The exact source data in [F1] fix the large qcqs site, the derived localization and descent of $B_{j}$ . Their adjunction gives $Map (L_{c d h} τ^{\leq j} B_{j}, B_{j}) ≃ Map (τ^{\leq j} B_{j}, B_{j}),$ where maps on the left are between cdh-local derived functors and maps on the right are presheaf maps. Take the image of the actual truncation map under this equivalence. This constructs $ϵ_{j}$ with the stated unit equation and a contractibly specified choice in the source derived enhancement.

2.1F1step 1.1∎

Evaluate this natural transformation at any source scheme. Its source is the evaluated sheafified functor, not the sheafification of an isolated complex at a point. All restriction squares commute because the adjunction was applied to functors. No bound on the cofiber follows merely from this adjunction; that bound will be imported separately.

Depends on

Exact cdh factorization inputs for the source étale map

Used by · 2

Dependency tree · 2 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.