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.
Composition series and composition factors of an object
Definition
A composition series of an object in an abelian category is a finite strict chain of subobjects
such that every quotient object is simple (Simple object, The quotient of an object by a subobject).
The simple quotient objects are the composition factors of the series.
Depends on
Used by
- Object of finite length Definition
- Truncation at a finite downward-closed ideal of a linkage class Definition
- Type of a module with a Verma filtration Definition
- A nonsplit simple has Hom-pairing diagonal two Example
- FALSE: every object of an abelian category has a composition series False statement
- Composition factors of the BGG kernel lie above the degree (BGG 10.6a) Lemma
- Fitting decomposition in a finite-length abelian category Lemma
- Hom from a projective counts simple composition factors Lemma
- Jordan-Holder factors of Verma modules dominate the head (BGG 8.12) Lemma
- Nonzero highest-weight images survive modulo n-minus (BGG 10.6b) Lemma
- Projective epimorphisms onto the simples generate every finite-length object Lemma
- Weight-lambda vectors are singular at a maximal label Lemma
- Finite-dimensional module categories satisfy the intrinsic finiteness conditions Proposition
- Category O has enough projectives Theorem
- Intrinsic finite category hypotheses give a finite projective generator Theorem
- Jordan-Holder theorem in an abelian category Theorem
- Projective and simple classes are dual bases under splitting Theorem
- Schreier refinement theorem in an abelian category Theorem
- Shift-orbit bases for graded simple and projective classes Theorem
- Simple classes freely generate the Grothendieck group of a length category Theorem
- Zassenhaus butterfly lemma in an abelian category Theorem
Dependency tree · two levels
7 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
- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, Section 1.5 (standard reference, not scraped)