CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-01

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 chain map between acyclic complexes has an acyclic cone

Statement

If $f : C_{∙} \to D_{∙}$ is a chain map and both $C_{∙}$ and $D_{∙}$ are acyclic, then $Cone (f)$ is acyclic.

Facts & Assumptions

Given: A chain map $f : C_{∙} \to D_{∙}$ between acyclic complexes.

[L1]

The cone long exact sequence exists (The cone long exact sequence).

[L2]

Acyclic means vanishing homology in every degree (Exactness of a complex at a degree and acyclic complexes).

Proof

technique · direct

1.1L1L2givenalgebra

In the cone long exact sequence from [L1], every copy of $H_{n} (C)$ and $H_{n} (D)$ is zero by [L2].

2.1L1L2step 1.1algebra∎

Exactness then forces every $H_{n} (Cone (f))$ to vanish as well. Hence the cone is acyclic.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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)