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.
FALSE: this page proves the diagram lemmas by reducing to the module case
Statement
The proof of the diagram lemmas supplied on this page reduces the connecting-morphism construction to the already-published module case via Freyd-Mitchell.
Facts & Assumptions
Given: The embedding-theorem route just described.
The connecting morphism is constructed intrinsically on this page (The connecting morphism exists and is unique).
Refutation
The construction in [L1] takes place in the given abelian category: it forms the connecting morphism through its intrinsic kernels and cokernels and proves the required factorization and uniqueness there. This is the proof supplied on the page for the connecting-morphism step in the diagram lemmas.
Consequently the statement, read as a description of this page's proof, is false: the connecting-morphism argument it supplies is the intrinsic argument in step 1.1. The assertion here concerns the displayed proof method; it makes no claim that an embedding argument is mathematically impossible in a setting where an appropriate embedding has been established.
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
- Charles A. Weibel, An Introduction to Homological Algebra, Section 1.3 (standard reference, not scraped)
- Peter Freyd, Abelian Categories (standard reference, not scraped)