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

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The long exact sequence of relative homology for a composable pair

Statement

For composable chain maps in an abelian category $C_{∙} \overset{f}{\to} D_{∙} \overset{g}{\to} E_{∙},$ there is an exact sequence $\dots \to H_{n} (D, C; f) \to H_{n} (E, C; g f) \to H_{n} (E, D; g) \to H_{n - 1} (D, C; f) \to \dots .$

Facts & Assumptions

Given: Composable chain maps $C_{∙} \overset{f}{\to} D_{∙} \overset{g}{\to} E_{∙}$ in an abelian category.

[L1]

Relative homology of a chain map is the homology of its mapping cone (The relative homology of a chain map).

[L2]

Every chain map has a cone long exact sequence (The cone long exact sequence).

[L3]

For the induced map $α : Cone (f) \to Cone (g f)$ , the cone $Cone (α)$ is chain-homotopy equivalent to $Cone (g)$ (The three-cone calculation for a composite chain map).

[L4]

Every chain-homotopy equivalence is a quasi-isomorphism (A chain homotopy equivalence is a quasi-isomorphism).

[L5]

Every chain map induces a well-defined map on homology (A chain map induces a well-defined map on homology).

Proof

technique · direct

1.1L2givenconstruct

Apply [L2] to the induced chain map $α : Cone (f) \to Cone (g f)$ . This gives an exact sequence $\dots \to H_{n} (Cone (f)) \to H_{n} (Cone (g f)) \to H_{n} (Cone (α)) \to H_{n - 1} (Cone (f)) \to \dots .$

2.1L1L3L4L5step 1.1algebra∎

Rewrite the first two terms by [L1]. By [L3] there is a chain-homotopy equivalence $Φ : Cone (α) \to Cone (g)$ ; [L4] makes $Φ$ a quasi-isomorphism, and [L5] therefore gives isomorphisms on homology. So $H_{n} (Cone (α)) ≅ H_{n} (Cone (g))$ , and rewriting the last term with [L1] yields exactly the displayed long exact sequence of relative homology for the composable pair.

Depends on

Used by

Relative homology of a composable pair of stalk complexes Example

Dependency tree · two levels

20 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)