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.

Split short exact sequence in an abelian category

Definition

A short exact sequence $0 \to A \overset{i}{\to} B \overset{p}{\to} C \to 0$ in an abelian category is split when there exist morphisms $s : C \to B, π : B \to A$ such that $p s = 1_{C}, π i = 1_{A}, i π + s p = 1_{B} .$

Equivalently, the five-tuple $(B, i, s, π, p)$ is the biproduct diagram of $A$ and $C$ in the sense of Biproduct. The point of the definition is the displayed compatible structure, not merely an abstract isomorphism $B ≅ A \oplus C$ .

Split short exact sequence in an abelian category

Definition

Depends on

Dependency tree · 2 levels

Sources