Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated
How 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.

Coefficient-ideal control with centres allowed off the subvariety

Statement

Assume AC (The Axiom of Choice), μ≥1, and either char⁡K=0 or K perfect with char⁡K=p>μ (Field). In the setting of The coefficient ideal controls the support after restriction, let (Xi) be a multiple test blow-up of (I,μ) whose centers Ci are either contained in the strict transforms Si of S or disjoint from them, for every i. Then the restrictions σi∣Si define a multiple test blow-up (Si) of C(I,μ)∣S and supp⁡(Ii,μ)∩Si=supp⁡[C(I,μ)∣S]i for every i.

Facts & Assumptions

Given: Assume AC. Let K be a field and let μ≥1 with char⁡K=0 or K perfect with char⁡K=p>μ; let (I,E,μ) be a marked ideal of maximal order whose support does not contain a smooth subvariety S⊆X having SNC with E; and let (Xi) be a multiple test blow-up all of whose centers Ci are either contained in the strict transform Si of S or disjoint from it.

[A1]

The Axiom of Choice: AC is used through the restriction-support supplier [F1].

[F1]

The coefficient ideal controls the support after restriction, Field: under AC and the stated characteristic bound, for centers contained in the strict transforms, (Si) is a multiple test blow-up of C(I,μ)∣S and supp⁡(Ii,μ)∩Si=supp⁡[C(I,μ)∣S]i.

[F2]

Restriction of a marked ideal to a smooth subvariety and its blow-ups: the restriction of a controlled transform is the controlled transform of the restriction along the strict transform, and the restriction of the marked ideal does not change when the blow-up center is disjoint from S.

[F3]

Multiple test blow-ups, controlled transforms and resolutions of marked ideals: a blow-up with center disjoint from Si restricts to an isomorphism over Si and does not modify the restricted marked ideal.

Proof

1.1A1F1F2F3

The restriction sequence is a multiple test blow-up. At each step, if Ci⊆Si then the restricted center is admissible for C(I,μ)∣S and the transform rule is [F2]; if Ci∩Si=∅ then the step restricts to an isomorphism over Si and does not change the restricted marked ideal by [F3]. Hence the restrictions of the morphisms define a multiple test blow-up (with isomorphism steps allowed) of C(I,μ)∣S.

2.1A1F1F2F3step 1.1∎

Equality of supports persists. For steps with centers contained in Si the equality is [F1]; for steps with centers disjoint from Si, both sides are unchanged: supp⁡(Ii+1,μ)∩Si+1=supp⁡(Ii,μ)∩Si because the blow-up is an isomorphism near Si and the strict transform of S is identified with S, and supp⁡[C(I,μ)∣S]i+1=supp⁡[C(I,μ)∣S]i by [F3]. Induction over the steps gives the asserted identity at every stage.

Depends on

Used by

Dependency tree · two levels

28 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