DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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 extensions

Definition

For extensions $ξ : 0 \to N \to E \to M \to 0$ and $ξ^{'} : 0 \to N \to E^{'} \to M \to 0$ , an equivalence is a morphism $u : E \to E^{'}$ making the diagram of short exact sequences commute with identity maps on $N$ and $M$ . Equivalence is deliberately stronger data than merely an isomorphism $E ≅ E^{'}$ : the maps to the fixed endpoints must be respected.

Depends on

An extension of an object by an object in an abelian category

Used by

Equivalent extensions have the same Ext class Lemma
A morphism of extensions is an isomorphism Proposition
Equivalence of extensions is an equivalence relation Proposition

Dependency tree · two levels

2 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, An Introduction to Homological Algebra, Chapters 3–4 (standard reference, not scraped)