DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-01‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

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.

Depends on

Used by

The homological delta-functor carried by homology of complexes Definition
Exactness at the homology of the left complex Lemma
Exactness at the homology of the right complex Lemma
Exactness at the target of the connecting map Lemma
Elementwise formula for the connecting map in module categories Proposition
The connecting morphism vanishes for a chain-split short exact sequence Proposition
Naturality of the homology connecting morphism Theorem
The long exact sequence in homology Theorem

Dependency tree · two levels

8 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, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 12.13: Complexes (standard reference, not scraped)