Alphabeta MathHow 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.
Abelian subcategory and exact embedding
Definition
Let be a full subcategory of an abelian category . The subcategory is an abelian subcategory when for every morphism of its kernel and cokernel computed in again lie in , and when is closed under finite biproducts in .
An exact embedding is a full embedding (Embedding and full embedding of categories) whose essential image is an abelian subcategory and whose underlying functor is additive (Additive functor).
Depends on
Used by
- Finite-support families of finite-dimensional vector spaces are locally finite but not finite Lemma
- Finite-dimensional module categories satisfy the intrinsic finiteness conditions Proposition
- Kernels and cokernels of quasi-coherent modules Theorem
- Objects of finite length form an abelian subcategory Theorem
- Quasi-coherence of pushforward for qcqs morphisms 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
- Barry Mitchell, Theory of Categories, Chapter I (standard reference, not scraped)
- Peter Freyd, Abelian Categories, Chapter 7 (standard reference, not scraped)