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

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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 connecting map for left derived functors Definition
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
The exact couple and subquotient constructions of the filtered complex spectral sequence agree 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)