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.
Irreducible components as schemes
Definition
An irreducible component of a scheme is a maximal irreducible closed subset of , equipped, when regarded as a scheme, with its reduced induced closed subscheme structure.
Depends on
Used by
- Locally factorial scheme Definition
- Weil divisor normal noetherian scheme Definition
- A regular point lies on one irreducible component Lemma
- Closed finite-index subgroups of rational points of smooth connected groups over algebraically closed fields are the whole point group Lemma
- Cycles of coherent sheaves and of closed subschemes, with flat pullback Lemma
- Dense relative-dimension strata in flat finitely presented fibres Lemma
- Morphisms from complete connected schemes to affine schemes are constant Lemma
- SNC projective compactifications of smooth quasiprojective varieties Lemma
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, Schemes, Section 11 (standard reference, not scraped)