CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Null-homotopic maps induce zero on homology

Statement

If $f : C_{∙} \to D_{∙}$ is null-homotopic, then $H_{n} (f) = 0 : H_{n} (C) \to H_{n} (D)$ for every $n \in Z$ .

Facts & Assumptions

Given: A null-homotopic chain map $f : C_{∙} \to D_{∙}$ and an integer $n$ .

[L1]

A null-homotopic map is chain homotopic to the zero chain map (A null-homotopic chain map).

[L2]

Chain-homotopic maps induce the same homology map (Chain-homotopic maps induce the same map on homology).

Proof

technique · direct

1.1

By [L1], the map $f$ is homotopic to $0 : C_{∙} \to D_{∙}$ .

L1given

2.1

Applying [L2] to the homotopy from step 1.1 gives $H_{n} (f) = H_{n} (0)$ . The map $H_{n} (0)$ is the zero morphism, so $H_{n} (f) = 0$ .

L2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)