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.

Prime-field restriction and the intrinsic étale coefficient complex

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. Let $k$ have prime subfield $F$ , let $X / k$ be qcqs, $j \geq 0$ , and let $n \geq 1$ be invertible in $k$ . Then $X$ is a qcqs $F$ -scheme, $n$ is invertible in $F$ , and the completed motivic complex at $X$ is the actual source value $M_{j} (X) = Z (j)_{m o t, F} (X)$ . Put $B_{j} (X) = R Γ_{e t} (X, μ_{n}^{\otimes j})$ , using the intrinsic small étale site of $X$ and the specified tensor powers over $Z / n$ . Forgetting the k-structure changes neither this site nor its coefficient sheaf or derived evaluation. Thus the source complex $B_{j} (X)$ is exactly the completed local étale target, with its pullbacks.

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 the fixed size bounds are retained for the K-injective derived-evaluation interface invoked in step 2.1. The Axiom of Choice

[F1]

Motivic complexes and cohomology fixes prime-field restriction of the motivic functor, and Roots of unity and étale Tate twists fixes the intrinsic small étale site and tensor powers.

[F2]

Under [A1], Derived global sections on a scheme site computes derived evaluation by a K-injective resolution on that chosen small site.

Proof

1.1F1

Quasi-compactness and quasi-separatedness belong to the underlying scheme, so forgetting to $F$ gives a source object. If $char k = p > 0$ , invertibility of $n$ means $p ∤ n$ , hence invertibility in $F = F_{p}$ ; if the characteristic is zero, every positive integer is invertible in $F = Q$ . The motivic identification is the defining precomposition of [F1].

2.1A1F1F2step 1.1∎

An object of the small étale site is an étale morphism $U \to X$ , and its covers and morphisms are defined over $X$ . This category, its topology and the equation $u^{n} = 1$ on its objects do not mention the presentation of $X$ over $k$ or $F$ . The tensor powers, including the zeroth power $Z / n$ , are therefore the same sheaves. Retain the same size-bounded small site in both presentations. Under [A1], apply the K-injective derived-evaluation interface of [F2] to this same site and sheaf; the resulting derived-category value is independent of the resolution. Pullback along a $k$ -morphism uses the same underlying morphism and coefficient maps after forgetting. This proves the claim about actual functors, not only abstract group isomorphisms.

Depends on

Used by · 2

Dependency tree · 3 levels

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