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.

Cofiber range gives natural truncation and cohomology comparison

PDF p. 5: Prime-to-characteristic comparison

Statement

Assume the Axiom of Choice and compatible fixed universe size bounds for the small sites and derived categories used below. For every qcqs $k$ -scheme $X$ , $j \geq 0$ and $n \geq 1$ invertible in $k$ , the actual comparison map gives a natural equivalence $τ^{\leq j} (Z (j)_{m o t} (X) / n) \tilde{\to} τ^{\leq j} R Γ_{e t} (X, μ_{n}^{\otimes j})$ and natural isomorphisms $H_{m o t}^{a, j} (X, Z / n) \to H_{e t}^{a} (X, μ_{n}^{\otimes j})$ for all $a \leq j$ .

Facts & Assumptions

Given: The hypotheses in the statement.

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

[A1]

The Axiom of Choice and fixed size bounds are inherited when applying the prime-field helper in steps 1.1 and 2.1, specifically its K-injective derived-evaluation identification. The Axiom of Choice

[F1]

The actual composite and its objectwise cofiber supplies the actual source comparison and its objectwise cofiber bound.

[F2]

Prime-field restriction and the intrinsic étale coefficient complex identifies the underlying prime-field object and the intrinsic étale target under [A1]; Motivic complexes and cohomology fixes its motivic source and finite coefficients.

Proof

1.1A1F1F2

Apply the source comparison of [F1] at the underlying prime-field object given by [F2]. Precomposition evaluates the same actual map and its same cofiber. Its triangle $A \overset{f}{\to} B \to C$ has $H^{a} (C) = 0$ for $a \leq j$ . For $a \leq j$ , both $H^{a - 1} (C)$ and $H^{a} (C)$ are zero, so the long exact sequence shows $H^{a} (f)$ is an isomorphism.

2.1A1F1F2step 1.1∎

Standard cohomological truncation preserves these cohomology groups in degrees at most $j$ and makes all larger ones zero. Hence $τ^{\leq j} f$ is a quasi-isomorphism and therefore an equivalence in $D (Z)$ . The coefficient and derived-evaluation identifications of [F2] identify its terms with the displayed ones. Applying cohomology gives the asserted degree- $a$ isomorphisms by the completed motivic definition. All steps use functors or the natural source triangle, so commute with pullback in $X$ .

Depends on

Used by · 1

Prime-to-characteristic étale comparison

Dependency tree · 3 levels

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