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.

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

Definition

An extension of $M$ by $N$ is a short exact sequence $0 ⟶ N \overset{i}{\to} E \overset{p}{\to} M ⟶ 0.$ Thus $N$ is the subobject and $M$ the quotient. This order is fixed throughout: such an extension will represent a class in ${Ext}^{1} (M, N)$ , not in ${Ext}^{1} (N, M)$ .

Depends on

Exact sequence and short exact sequence in an abelian category

Used by

An n-fold Yoneda extension Definition
Equivalence of extensions Definition
Pullback and pushout of an extension Definition
The derived Ext-one class of an extension Definition
The split extension class Definition
FALSE: an extension of M by N represents Ext one of N by M False statement

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)