TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 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 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

1.1

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

L2givenconstruct

2.1

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.

L1L3L4L5step 1.1algebra∎

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)