CounterexampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27‡ 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.

An infinite coproduct need not agree with the infinite product

Statement refuted

Refuted claim: in an additive category, an infinite coproduct is always the same object as the corresponding infinite product.

The witness is the countable family of copies of $Z$ in $A b = Z - M o d$ .

Facts & Assumptions

Given: The countable family $(Z)_{n \in N}$ in $A b$ .

[L1]

The direct sum $⨁_{n \in N} Z$ consists of those integer sequences with finite support (The direct sum of an indexed family of modules).

[L2]

For every ring $R$ , the category $R - M o d$ has all small products and coproducts (For every ring R, the category R-Mod is complete and cocomplete).

Counterexample

technique · direct

1.1

By [L2], both $⨁_{n \in N} Z$ and $\prod_{n \in N} Z$ exist in $A b$ . The canonical map from the coproduct to the product sends a finitely supported sequence to the same sequence viewed in the full product.

L1L2

2.1

The product element $(1, 1, 1, \dots)$ is not in the image of that map, because every element of the direct sum has finite support by [L1]. Therefore the canonical map is not surjective, hence not an isomorphism.

L1step 1.1

3.1

So an infinite coproduct need not agree with the corresponding infinite product.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Peter Freyd, Abelian Categories, Exercise 2A (standard reference, not scraped)
A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.14 and 3.15 (standard reference, not scraped)