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.

Equivalence of the weight-zero pullback projection

PDF p. 4: Weight zero

Statement

The canonical source map $ρ_{0} : Z (0)_{m o t, F} \to Z (0)_{c d h, F}$ is an equivalence on every qcqs $F$ -scheme, and hence after restriction to qcqs $k$ -schemes. This uses its actual pullback construction, rather than choosing equivalences from a comparison of groups.

Facts & Assumptions

Given: The hypotheses in the statement.

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

[F1]

Exact source weight-zero and constant-sheaf input imports the cartesian squares and right equivalences in EM Examples4.11 and4.26, with their actual maps.

Proof

1.1F1construct

The exact source input [F1] gives the source cartesian square with left projection $ρ_{0}$ and an equivalence on the right vertical arrow. A base change of an equivalence is an equivalence: the pullback of $B \tilde{\to} D \leftarrow C$ maps equivalently to $C$ , since replacing $B$ by $D$ identifies the pullback with $C$ . Apply this in the stable derived category to that square.

2.1F1step 1.1∎

This gives precisely its left projection $ρ_{0}$ , naturally in the source object. Restrict this actual map by Prime-field restriction preserves actual comparison diagrams. The same equivalence holds at each qcqs $k$ -scheme.

Depends on

Used by · 2

Dependency tree · 3 levels

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