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.

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Evaluation preserves chain homotopies

PDF p. 4: Derived Nisnevich and cdh sections

Statement

For an object $U$ of a small site, evaluation sends complexes of abelian sheaves and chain maps to complexes and chain maps of abelian groups, and sends chain-homotopic maps to chain-homotopic maps. In particular it sends a chain-homotopy equivalence to a chain-homotopy equivalence.

A cochain complex has maps $d^{n} : C^{n} \to C^{n + 1}$ with $d^{n + 1} d^{n} = 0$ . A chain homotopy from $f$ to $g$ consists of maps $h^{n} : C^{n} \to D^{n - 1}$ satisfying $f - g = d h + h d$ . A homotopy equivalence has a chain-map inverse up to such homotopies.

Facts & Assumptions

Given: An object $U$ and complexes and maps as in the statement.

Goal: Prove preservation of complexes, homotopies and homotopy equivalences.

[F1]

Evaluation is additive. Evaluation preserves kernels

Proof

technique · Evaluate the defining identities

1.1F1given

Evaluation preserves composition, identities, sums and zero maps. Thus it carries $d^{2} = 0$ and the chain-map equations to the corresponding equations of abelian groups.

2.1F1step 1.1∎

Evaluating $f - g = d h + h d$ gives the homotopy identity for $h (U)$ . Evaluating the two inverse-up-to-homotopy identities for a homotopy equivalence gives the final assertion.

Depends on

Evaluation preserves kernels

Used by · 1

Negative cohomology of a nonnegative evaluated complex

Dependency tree · 3 levels

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