CorollaryStatement: 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.

Two-out-of-three for acyclicity in a short exact sequence of complexes

Statement

In a short exact sequence of complexes $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0,$ if any two of $A_{∙}$ , $B_{∙}$ , and $C_{∙}$ are acyclic, then so is the third.

Facts & Assumptions

Given: A short exact sequence $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0$ of complexes.

[L1]

The sequence carries a long exact sequence in homology (The long exact sequence in homology).

[L2]

Acyclic means vanishing homology in every degree (Exactness of a complex at a degree and acyclic complexes).

Proof

technique · direct

1.1

If $A_{∙}$ and $C_{∙}$ are acyclic, then for every $n$ the exact window $H_{n} (A) \to H_{n} (B) \to H_{n} (C)$ has zero outer terms by [L2]. Exactness from [L1] therefore gives $H_{n} (B) = 0$ for all $n$ .

L1L2givenalgebra

1.2

If $A_{∙}$ and $B_{∙}$ are acyclic, then the exact window $H_{n} (B) \to H_{n} (C) \to H_{n - 1} (A)$ has zero outer terms, so $H_{n} (C) = 0$ for all $n$ .

L1L2givenalgebra

2.1

If $B_{∙}$ and $C_{∙}$ are acyclic, then the exact window $H_{n} (C) \to H_{n - 1} (A) \to H_{n - 1} (B)$ has zero outer terms, so $H_{n - 1} (A) = 0$ for all $n$ . Thus the remaining complex is acyclic in every case.

L1L2givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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)