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 splitting lemma instantiated at the published module theorem
Example
The categorical splitting lemma Splitting lemma in an abelian category specializes in to the published module statement The splitting lemma for short exact sequences of modules. The section, retraction, and direct-sum identity are literally the same equations.
Facts & Assumptions
Given: A short exact sequence of modules together with either a section of its quotient map or a retraction of its inclusion.
From either a section or a retraction, the categorical splitting lemma produces the unique complementary retraction or section (Splitting lemma in an abelian category).
The published module splitting lemma states the same criterion in the module category (The splitting lemma for short exact sequences of modules).
Verification
A short exact sequence of modules is a short exact sequence in an abelian category, and the given section or retraction is exactly the additional datum required by [L1].
The complementary map produced by [L1] satisfies , , and , exactly the module equations stated in [L2].
Therefore the module theorem is the concrete instance of the categorical one.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
9 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
- The Stacks Project, Section 12.5, Lemma 12.5.10 (standard reference, not scraped)
- P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)