Alphabeta MathHow statement and proof provenance work
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Finite k-linear abelian categories
Definition
A finite -linear abelian category is a locally finite -linear abelian category with finitely many isomorphism classes of simple objects and enough projectives: every simple object has a projective cover. This is an intrinsic finiteness condition; it does not assert semisimplicity.
Depends on
Used by
- Finite length and finite Hom do not imply a finite category Counterexample
- Fusion and multifusion categories Definition
- Left and right Nakayama functors by finite kernel calculus Definition
- Superfluous subobjects and projective covers in an abelian category Definition
- The Deligne product of finite linear categories Definition
- Every finite k-linear abelian category is semisimple False statement
- Finite-support families of finite-dimensional vector spaces are locally finite but not finite Lemma
- Projective epimorphisms onto the simples generate every finite-length object Lemma
- Finite-dimensional module categories satisfy the intrinsic finiteness conditions Proposition
- Finite abelian categories admit finite-dimensional module models Theorem
- Finite Deligne products exist via tensor-product algebras Theorem
- Intrinsic finite category hypotheses give a finite projective generator Theorem
Dependency tree · two levels
8 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, Definitions 1.8.5–1.8.6 (standard reference, not scraped)