DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-01

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.

The connecting morphism in homology

Definition

Fix a short exact sequence of complexes in an abelian category, $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0,$ and an integer $n$ . Let $q_{n} : Z_{n} (C) \to H_{n} (C)$ be the homology quotient of Homology object of a chain complex. By The preconnecting arrow annihilates boundaries, the preconnecting arrow ${\tilde{\partial}}_{n} : Z_{n} (C) \to H_{n - 1} (A)$ kills the boundary subobject $B_{n} (C)$ . Therefore the cokernel property of $q_{n}$ gives a unique morphism $\partial_{n} : H_{n} (C) \to H_{n - 1} (A)$ such that $\partial_{n} q_{n} = {\tilde{\partial}}_{n} .$

This morphism is the connecting morphism in homology attached to the short exact sequence of complexes.

The connecting morphism in homology

Definition

Depends on

Dependency tree · 2 levels

Sources