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 long exact sequence in cohomology

Statement

Let $0 \to A^{∙} \to B^{∙} \to C^{∙} \to 0$ be a short exact sequence of cochain complexes in an abelian category. Then there is a natural exact sequence $\dots \to H^{n} (A) \to H^{n} (B) \to H^{n} (C) \overset{\partial^{n}}{\to} H^{n + 1} (A) \to H^{n + 1} (B) \to H^{n + 1} (C) \to \dots .$

Facts & Assumptions

Given: A short exact sequence $0 \to A^{∙} \to B^{∙} \to C^{∙} \to 0$ of cochain complexes.

[L1]

A cochain complex may be read as a reindexed chain complex with $(C^{♯})_{n} = C^{- n}$ (Cochain complex in an abelian category).

[L2]

The cohomology object $H^{n} (C)$ is the homology of that reindexed chain complex in degree $- n$ (Cohomology object of a cochain complex).

[L3]

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

Proof

technique · direct

1.1

By [L1], the given short exact sequence of cochain complexes is the same data as a short exact sequence of reindexed chain complexes. Applying [L3] gives a long exact sequence in homology for those chain complexes.

L1L3givenconstruct

2.1

Replace each term $H_{- n} (X^{♯})$ by $H^{n} (X)$ using [L2]. Under the same reindexing, the connecting map from degree $- n$ homology to degree $- n - 1$ homology becomes a map $\partial^{n} : H^{n} (C) \to H^{n + 1} (A) .$ This is the displayed long exact cohomology sequence.

L1L2step 1.1algebra∎

Depends on

Used by

A six-term cohomology sequence Example
FALSE: the cohomology connecting morphism lowers degree False statement
Naturality of the cohomology connecting morphism Theorem

Dependency tree · two levels

13 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)