DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-29

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.

Exact sequence and short exact sequence in an abelian category

Definition

A sequence of morphisms in an abelian category is exact when it is exact at every interior object in the sense of Exactness at a node.

In particular, a composable triple $0 \to A \overset{i}{\to} B \overset{p}{\to} C \to 0$ is a short exact sequence when it is exact at $A$ , at $B$ , and at $C$ .

This page uses the sequence language literally: the definition names exactness of displayed chains of morphisms, not chain complexes as objects.

Depends on

Exactness at a node

Exact sequence and short exact sequence in an abelian category

Definition

Depends on

Dependency tree · 2 levels

Sources