CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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 quasi-isomorphisms in a short exact sequence diagram

Statement

Consider a morphism of short exact sequences of complexes $\begin{array}{ccccccccc} 0 & \to & A_{∙} & \to & B_{∙} & \to & C_{∙} & \to & 0 \\ ↓ a & ↓ b & ↓ c \\ 0 & \to & A_{∙}^{'} & \to & B_{∙}^{'} & \to & C_{∙}^{'} & \to & 0. \end{array}$ If any two of $a$ , $b$ , and $c$ are quasi-isomorphisms, then so is the third.

Facts & Assumptions

Given: A morphism of short exact sequences of complexes.

[L1]

Such a ladder induces a morphism between the associated long exact homology sequences (The long exact homology sequence is natural).

[L2]

In a morphism of long exact sequences, if the four surrounding comparison maps in a five-term window are isomorphisms, then the middle one is an isomorphism (Five lemma for a morphism of long exact sequences).

[L3]

A quasi-isomorphism is a chain map inducing isomorphisms on all homology objects (Quasi-isomorphism).

Proof

technique · direct

1.1

Assume $b$ and $c$ are quasi-isomorphisms. In the long exact ladder from [L1], center the five-term window at $H_{n} (A)$ . The four surrounding comparison maps come from $H_{n + 1} (b)$ , $H_{n + 1} (c)$ , $H_{n} (b)$ , and $H_{n} (c)$ , so they are isomorphisms by [L3]. Hence $H_{n} (a)$ is an isomorphism by [L2].

L1L2L3givenalgebra

1.2

Assume $a$ and $c$ are quasi-isomorphisms. Center the five-term window at $H_{n} (B)$ . The surrounding comparison maps come from $H_{n + 1} (c)$ , $H_{n} (a)$ , $H_{n} (c)$ , and $H_{n - 1} (a)$ , so [L2] gives that $H_{n} (b)$ is an isomorphism.

L1L2L3givenalgebra

2.1

Assume $a$ and $b$ are quasi-isomorphisms. Center the five-term window at $H_{n} (C)$ . The surrounding comparison maps come from $H_{n} (a)$ , $H_{n} (b)$ , $H_{n - 1} (a)$ , and $H_{n - 1} (b)$ , so [L2] yields that $H_{n} (c)$ is an isomorphism. By [L3], the missing map is therefore a quasi-isomorphism in every case.

L1L2L3givenalgebra∎

Depends on

Used by

Short five lemma for quasi-isomorphisms Corollary
Two-out-of-three for a diagram of finite complexes Example

Dependency tree · two levels

8 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)