PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-01

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Homology of complexes satisfies the delta-functor naturality and exactness laws

Statement

For every abelian category, the family of homology functors together with the connecting morphisms of this page is a homological $δ$ -functor: it sends each short exact sequence of complexes to a long exact sequence, and it is natural under morphisms of short exact sequences.

Facts & Assumptions

Given: An abelian category and a short exact sequence of complexes in it.

[L1]

This page defines the family $(H_{n}, \partial_{n})$ as a concrete homological $δ$ -functor candidate (The homological delta-functor carried by homology of complexes).

[L2]

Short exact sequences of complexes carry long exact homology sequences (The long exact sequence in homology).

[L3]

Those long exact sequences are natural under morphisms of short exact sequences (The long exact homology sequence is natural).

Proof

technique · direct

1.1L1given

By [L1], the only axioms left to check are exactness for each short exact sequence and naturality for each morphism of such sequences.

2.1L2L3step 1.1algebra∎

Exactness is exactly [L2], and naturality is exactly [L3]. Therefore the family of homology functors with these connecting morphisms satisfies the required $δ$ -functor laws.

Depends on

Used by

A nonnatural choice of connecting maps does not form a delta functor Counterexample
Homology as a homological delta functor Example
FALSE: any sequence of functors with long exact sequences is a delta functor False statement

Dependency tree · two levels

11 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)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)