False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

FALSE: a generator is automatically projective

Statement

A generator is the same thing as a projective generator.

Facts & Assumptions

Given: The abelian group $G = Z \oplus Z / p$ for a prime $p$ .

[L1]

Generators are defined by separation of morphisms (Generator and cogenerator of a category).

[L2]

Direct summands of projectives are projective (A direct summand of a projective is projective).

[L3]

Projective objects are the lifting objects of Projective object.

Refutation

1.1L1algebra

The summand $Z$ is a generator of $A b$ , so $G = Z \oplus Z / p$ is also a generator: precompose with the inclusion $Z \to G$ and then use the generator property of $Z$ .

2.1L2L3step 1.1∎

If $G$ were projective, then its direct summand $Z / p$ would be projective by [L2]. But the quotient map $Z ↠ Z / p$ does not split, so $Z / p$ does not have the lifting property [L3]. Therefore $G$ is a generator that is not projective.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, Section 1.6 (standard reference, not scraped)