False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-30‡ 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.

FALSE: every infinite coproduct of complexes has homology equal to the coproduct of their homologies

Statement

In every abelian category, whenever an infinite coproduct of chain complexes exists, its homology is the coproduct of the homologies.

Facts & Assumptions

Given: An abelian category with small coproducts and a family of monomorphisms $u_{i} : A_{i} \to B_{i}$ .

[A1]

There exist abelian categories with small coproducts that do not satisfy AB4.

[L1]

AB4 means that every small coproduct of monomorphisms is again monic (The axioms AB4 and AB4*).

[L2]

Degreewise coproducts of complexes, when they exist, are computed degreewise (Products and coproducts of complexes are degreewise when they exist and preserve differentials).

[L3]

A complex is exact at degree $n$ exactly when its $n$ th homology is zero (A complex is exact at n exactly when its nth homology is zero).

[L4]

In an abelian category, a morphism is monic exactly when its kernel is zero (In an abelian category, monic means zero kernel and epic means zero cokernel).

Refutation

technique · direct

1.1

For each $i$ , form the two-term complex $C_{∙}^{(i)} : 0 \to A_{i} \overset{u_{i}}{\to} B_{i} \to 0,$ with $A_{i}$ in degree $1$ and $B_{i}$ in degree $0$ . Because $u_{i}$ is monic, [L4] gives $\ker (u_{i}) = 0$ , so [L3] implies $H_{1} (C^{(i)}) = 0$ for every $i$ .

L3L4given

2.1

If the statement were true, then by [L2] the coproduct complex $∐_{i} C^{(i)}$ would satisfy $H_{1} ⁣ (∐_{i} C^{(i)}) ≅ ∐_{i} H_{1} (C^{(i)}) = 0.$ Its degree- $1$ differential is exactly $∐_{i} u_{i}$ , so [L3] and [L4] would force $∐_{i} u_{i}$ to be monic. By [L1], that would make every abelian category with small coproducts satisfy AB4.

L1L2L3L4step 1.1

3.1

This contradicts [A1]. Therefore the statement is false.

A1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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, An Introduction to Homological Algebra, Appendix A.4 (standard reference, not scraped)