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.

A chosen root trivializes the actual Tate coefficient on the slice

PDF p. 6: Singular Beilinson–Lichtenbaum comparison

Statement

Let $k$ be a field, $p$ invertible in $k$ , and $ζ \in k$ of multiplicative order $p$ . On every small étale site of a qcqs $k$ -scheme $V$ the map $t_{ζ, V} : Z / p \to μ_{p}$ , $a \mapsto ζ^{a}$ , is an isomorphism of sheaves of $Z / p$ -modules. Its tensor cube gives $t_{ζ, V}^{(3)} : Z / p \to μ_{p}^{\otimes 3}$ , $1 \mapsto ζ^{\otimes 3}$ , naturally in $V$ . This natural transformation is on $C / U$ ; no such chosen-root map is asserted on all of $C$ .

Facts & Assumptions

[F1]

Roots of unity and étale Tate twists supplies the coefficient sheaves, tensor powers and factorization argument. The actual big cdh slice over a field fixes precisely the slice category. The constant coefficient is the constant étale sheaf, not a claim that its section group equals $Z / p$ on a disconnected scheme.

Proof

Given: $k, p, ζ, V$ as stated.

Goal: Construct the coefficient isomorphism with its exact naturality domain.

1.1F1given

The factorization $T^{p} - 1 = \prod_{a = 0}^{p - 1} (T - ζ^{a})$ has pairwise unit differences on every $k$ -scheme. A root on an affine étale object therefore gives a finite disjoint open decomposition on which it equals a unique $ζ^{a}$ . Thus the constant-sheaf map is locally bijective, and hence is a sheaf isomorphism, as in [F1].

2.1F1step 1.1∎

Tensor this map three times over the constant ring sheaf $Z / p$ and identify $(Z / p)^{\otimes 3}$ with $Z / p$ using multiplication. Tensoring the inverse gives its inverse. Every $k$ -morphism carries $ζ$ to the same chosen base-field section, so these isomorphisms commute with pullback. The root need not exist over an arbitrary object of the ambient site, and none is used there.

Depends on

Used by · 3

Dependency tree · 3 levels

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