False statementConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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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.
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Every finite k-linear abelian category is semisimple

Statement

False claim. Every finite $k$ -linear abelian category is semisimple.

Facts & Assumptions

Given: The two definitions on this page.

[F1]

Finiteness requires local finiteness, finitely many simple classes, and enough projectives (Finite k-linear abelian categories).

[F2]

Semisimplicity requires every object to be a direct sum of simples (Semisimple objects and semisimple abelian categories).

Refutation

technique · direct

1.1F1construct

Let $A = k [ε] / (ε^{2})$ and let $C = A - m o d$ be the category of finite-dimensional left $A$ -modules. It is finite: $A$ is finite-dimensional, it has the single simple module $S = A / (ε)$ , and the quotient $A ↠ S$ is its projective cover.

2.1step 1.1F2∎

The sequence $0 \to S \overset{1 \mapsto ε}{\to} A \to S \to 0$ does not split. Indeed, every vector killed by $ε$ in $A$ is a multiple of $ε$ and maps to zero in the quotient, so no section exists. Thus $A$ is not a direct sum of simples, and [F2] says that $C$ is not semisimple.

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

Etingof, Gelaki, Nikshych, Ostrik, Tensor Categories, Section 1.8 (standard reference, not scraped)