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.
Why scheme fibre products are needed beyond the classical setting
Classical affine products here use reduced coordinate algebras over an algebraically closed field and only construct selected pullbacks. Tensor products and quotients can retain nilpotents or acquire additional components after a field extension, data that a reduced point set discards. Scheme fibre products retain that data and provide the unrestricted existence theorem; they are developed later rather than silently imported here.
Depends on
Used by
Dependency tree · two levels
5 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
- J. S. Milne, Algebraic Geometry, Aside 5.34 (standard reference, not scraped)
- MIT 18.725 Algebraic Geometry, Lecture 7, Example 9 (standard reference, not scraped)