DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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

Used by

Split short exact sequence in an abelian category Definition
A short exact sequence is a kernel-cokernel pair Theorem
A square is cartesian exactly when a short sequence is exact Theorem
AB5 is equivalent to exactness of filtered colimits Theorem
Degenerate exactness criteria Theorem

Dependency tree · two levels

3 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

The Stacks Project, Section 12.5, Definition 12.5.7 (standard reference, not scraped)
David Mehrle, Category Theory, Part III, Definition 7.20 (standard reference, not scraped)