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.
Equivalence of marked ideals
Definition
Two marked ideals and on the same smooth -scheme are equivalent, if: (1) as ordered families of divisors; (2) their supports agree; and (3) the multiple test blow-ups of the one are exactly the multiple test blow-ups of the other, and for every such blow-up the induced supports agree, for every (Multiple test blow-ups, controlled transforms and resolutions of marked ideals). The relation is reflexive, symmetric and transitive by definition, and the algorithm below replaces a marked ideal by equivalent ones at the steps marked in the source.
Depends on
Used by
- Canonical resolutions with invariants of a marked ideal Definition
- The monomial part, the non-monomial part and the companion ideal Definition
- A marked ideal is equivalent to its powers Lemma
- Addition and multiplication of marked ideals Lemma
- Canonical resolution under isomorphisms of the ground field Lemma
- Canonical resolutions commute with embeddings of ambient smooth schemes Lemma
- Canonical resolutions over non-algebraically-closed ground fields Lemma
- Etale commutativity of the companion-ideal step Lemma
- Etale commutativity of the maximal-order resolution step Lemma
- The coefficient ideal is equivalent to the marked ideal Lemma
- The homogenized ideal is equivalent to the marked ideal Lemma
- Canonical resolution of marked ideals Proposition
- Canonical principalization of ideals in characteristic zero Theorem
Dependency tree · two levels
14 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.