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.
The quotient of an object by a subobject
Definition
Let be a subobject of an object in an abelian category, represented by a monomorphism . The quotient of by is the quotient object represented by the cokernel of :
Because every monomorphism is the kernel of its cokernel (Every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel), this quotient is the object paired with the given subobject by the kernel-cokernel correspondence. The next item checks that the definition does not depend on the chosen representative of the class .
Depends on
Used by
- Associated graded object of a filtered object Definition
- Composition series and composition factors of an object Definition
- The axioms AB5 and AB5* Definition
- 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
- The exact sequence for adding one point to a divisor Lemma
- Finite abelian categories admit finite-dimensional module models Theorem
- First isomorphism theorem in an abelian category Theorem
- Noether isomorphism theorems recovered from the nine lemma Theorem
- Simple classes freely generate the Grothendieck group of a length category Theorem
- The quotient by a subobject is independent of the chosen representing monomorphism Theorem
- Third isomorphism theorem in an abelian category Theorem
Dependency tree · two levels
12 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
- Junhan Tan, The Freyd-Mitchell Embedding Theorem, Theorem 2.3 (standard reference, not scraped)