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.
This page uses Grothendieck's AB1 and AB2 labels, and records the competing conventions
Grothendieck's original Tohoku paper labels the two extra clauses on top of additivity as AB1 and AB2: every morphism has a kernel and a cokernel, and the canonical map is an isomorphism. This page follows that convention.
Two cautions matter because both conventions occur in the modern literature. First, there is no Grothendieck axiom "AB0": additivity is a standing hypothesis, not a numbered clause. Second, Weibel's Appendix A uses the label "AB2" for a different statement, namely that every monomorphism is the kernel of its cokernel. That statement is proved later on this page as Every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel.
The Stacks Project avoids the AB labels altogether and writes the same content directly as the definition of an abelian category. That is compatible with this page's choice; it is a notation split, not a mathematical disagreement.
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Dependency tree · two levels
3 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
- Alexandre Grothendieck, Some aspects of homological algebra, §1.4 (standard reference, not scraped)
- Charles Weibel, An Introduction to Homological Algebra, Appendix A.4 (standard reference, not scraped)
- The Stacks Project, Section 12.5 (standard reference, not scraped)