PropositionStatement: Literature-sourcedProof: AI-generated

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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.

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Prime-to-characteristic étale comparison

PDF p. 5: Property (5)

Statement

Assume the Axiom of Choice and compatible fixed universe size bounds for the small sites and derived categories used below. Let $X$ be any qcqs scheme over $k$ , $j \geq 0$ , and $n \geq 1$ prime to the characteristic of $k$ (equivalently invertible in $k$ ). There is the natural source comparison $Z (j)_{m o t} (X) / n ⟶ R Γ_{e t} (X, μ_{n}^{\otimes j}),$ whose cofiber has cohomology only in degrees $> j$ . In particular it induces a natural equivalence $τ^{\leq j} (Z (j)_{m o t} (X) / n) ≃ τ^{\leq j} R Γ_{e t} (X, μ_{n}^{\otimes j})$ and isomorphisms $H_{m o t}^{a, j} (X, Z / n) ≅ H_{e t}^{a} (X, μ_{n}^{\otimes j})$ for $a \leq j$ . The quotient is derived, the tensor power uses the specified small étale coefficient sheaf, and the natural map is the composite through source cdh-local motivic cohomology fixed in the prerequisites.

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 in steps 1.1 and 2.1 from the prime-field and truncation helpers, through their selected K-injective derived-sections interface. The Axiom of Choice

[F1]

Prime-field restriction and the intrinsic étale coefficient complex supplies prime-field restriction and the unchanged intrinsic étale target under [A1].

[F2]

The exact cdh-local-to-objectwise étale map and The actual finite-quotient map to cdh-local motivic cohomology supply the two maps; The actual composite and its objectwise cofiber supplies their source composite and its imported objectwise cofiber bound.

[F3]

Cofiber range gives natural truncation and cohomology comparison supplies the natural truncation equivalence and cohomology comparison under [A1].

Proof

1.1A1F1F2

The first prerequisite places $X$ in the qcqs prime-field category without changing either the completed motivic functor value or the intrinsic étale coefficient complex. The next two supply the actual maps $ϵ_{j}$ and $α_{j}$ with their precise localization and finite-quotient meanings. The fourth forms their specified source composite and applies its imported objectwise cofiber bound. All source hypotheses hold by the first prerequisite.

2.1A1F3step 1.1∎

The fifth prerequisite applies the long exact sequence and cohomological truncation to this actual map, giving the displayed equivalence and all the asserted degree- $a$ isomorphisms with naturality. This proves exactly the comparison through weight $j$ on the full qcqs scope. In particular it asserts no degree-five weight-three comparison.

Depends on

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.

Sources

Fasel–Yakerson · p. 5, property (5)