Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28
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FALSE: every abelian category has a generator

Statement

Every abelian category has a generator.

Facts & Assumptions

Given: The abelian category FinAb of finite abelian groups.

[L1]

A generator must separate distinct morphisms by precomposition (Generator and cogenerator of a category).

[L2]

An abelian category is a category with the usual additive exact structure (Abelian category).

Refutation

1.1

The category FinAb is abelian: kernels, cokernels, and finite biproducts of morphisms of finite abelian groups are again finite abelian groups. So [L2] applies to it.

L2algebra
2.1

Let G be any finite abelian group. Choose a prime p not dividing the exponent of G. Then every homomorphism G→Z/p is zero. Hence the identity map and the zero map of Z/p cannot be separated by precomposition with any map from G, so G is not a generator by [L1]. Since G was arbitrary, FinAb has no generator.

L1step 1.1choose
3.1

Thus not every abelian category has a generator.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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