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CounterexampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)
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  • 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.

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Finite length and finite Hom do not imply a finite category

Statement refuted

Every k-linear abelian category with finite-dimensional hom-spaces, finite-length objects and enough projective covers is a finite k-linear abelian category.

Facts & Assumptions

Given: A field k and the category C of finite-support N-indexed families (Vn) of finite-dimensional k-vector spaces with componentwise linear maps.

[L1]

C is k-linear and abelian; every hom-space is finite-dimensional over k; every object has finite length and is projective; the objects Sm with (Sm)m=k and (Sm)n=0 for n≠m are pairwise non-isomorphic simple objects; and no object of C is a generator (Finite-support families of finite-dimensional vector spaces are locally finite but not finite).

[L2]

A projective cover of X is an essential epimorphism Q↠X with Q projective; an epimorphism is essential when its kernel is superfluous, and the zero subobject is superfluous because [0]∨[m]=[m] for every subobject [m] (Superfluous subobjects and projective covers in an abelian category).

[L3]

A locally finite k-linear abelian category has finite-dimensional hom-spaces and every object of finite length; a finite k-linear abelian category is such a category with finitely many isomorphism classes of simple objects and enough projectives, that is, a projective cover of every simple object (Locally finite k-linear abelian categories, Finite k-linear abelian categories, Simple object).

Counterexample

technique · direct
1.1L1L3given

By [L1] the category C is a k-linear abelian category with finite-dimensional hom-spaces and finite-length objects; equivalently it is a locally finite k-linear abelian category in the sense of [L3].

2.1L1L2step 1.1

Every simple object S of C has a projective cover: S is projective by [L1], so the identity 1S:S→S is an epimorphism with projective source whose kernel is the zero subobject, which is superfluous by [L2]; hence 1S is an essential epimorphism and a projective cover of S.

3.1L1L3step 2.1

Thus C satisfies all the hypotheses of the refuted statement: it is k-linear and abelian with finite-dimensional hom-spaces, all objects have finite length, and every simple object has a projective cover by step 2.1. But by [L1] its simple objects Sm are pairwise non-isomorphic, one for each m∈N, so C has infinitely many isomorphism classes of simple objects; by [L3] it is therefore not a finite k-linear abelian category. This refutes the statement.

4.1L1given∎

The failure is exactly the failure of the finite-simple-classes clause of the intrinsic definition while all the other clauses hold, and [L1] additionally supplies that no object of C is a generator; the construction of [L1] uses no choice, so no choice is used here.

Depends on

Used by

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Dependency tree · two levels

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Sources