TheoremStatement: Literature-sourcedProof: 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 cone long exact sequence

Statement

For every chain map $f : C_{∙} \to D_{∙}$ in an abelian category, there is an exact sequence $\dots \to H_{n} (C) \overset{H_{n} (f)}{\to} H_{n} (D) \to H_{n} (Cone (f)) \to H_{n - 1} (C) \overset{H_{n - 1} (f)}{\to} H_{n - 1} (D) \to \dots .$

Facts & Assumptions

Given: A chain map $f : C_{∙} \to D_{∙}$ .

[L1]

The canonical cone sequence $0 \to D_{∙} \to Cone (f)_{∙} \to C [1]_{∙} \to 0$ is degreewise split short exact (The canonical mapping-cone sequence is degreewise split short exact).

[L2]

Homology of a shift satisfies $H_{n} (C [1]) ≅ H_{n - 1} (C)$ (Homology of a shift is shifted homology).

[L3]

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

[L4]

Every short exact sequence of complexes yields a long exact homology sequence (The long exact sequence in homology).

[L5]

The cone differential on $D_{n} \oplus C_{n - 1}$ is $d (y, x) = (d_{D} y + f (x), - d_{C} x)$ (The mapping cone of a chain map).

[L6]

In the weaker snake construction, the connecting morphism is obtained from a pullback $P$ with maps $π : P \to \ker (γ)$ and $r : P \to U$ and is characterized by $δ π = q_{α} r$ (Snake lemma under the weaker Stacks hypotheses).

Proof

technique · direct

1.1

Apply [L4] to the short exact sequence from [L1]. This gives an exact sequence $\dots \to H_{n} (D) \to H_{n} (Cone (f)) \to H_{n} (C [1]) \overset{δ_{n}}{\to} H_{n - 1} (D) \to \dots .$

L1L4givenconstruct

2.1

Apply the weaker snake construction [L6] to the quotient-kernel diagram of the cone sequence from [L1] in degree $n + 1$ . Let $z_{C} : Z_{n} (C) \to C_{n}$ be the cycle inclusion and let $s : Z_{n} (C) \to Cone (f)_{n + 1}$ have components $(0, z_{C})$ . Its projection to $C [1]_{n + 1} = C_{n}$ is $z_{C}$ , so $s$ and the quotient $q_{C} : Z_{n} (C) \to H_{n} (C)$ induce a morphism $t : Z_{n} (C) \to P$ into the pullback used in [L6], with $π t = q_{C}$ . By [L5], $d^{Cone (f)} s = j_{n} f_{n} z_{C},$ and the defining equation for $r$ in the snake construction therefore gives $r t = Z_{n} (f)$ . Consequently [L6] yields $δ_{n + 1} q_{C} = δ_{n + 1} π t = q_{D} r t = q_{D} Z_{n} (f) = H_{n} (f) q_{C} .$ The last equality is the defining square for [L3]. Since $q_{C}$ is epic, $δ_{n + 1} = H_{n} (f)$ . Reindexing step 1.1 by [L2] gives the displayed cone long exact sequence.

L1L2L3L5L6step 1.1constructalgebra∎

Depends on

Used by

Dependency tree · two levels

29 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)