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.
Quasi-coherent ideal sheaves
Definition
Assume the Axiom of Choice (The Axiom of Choice), inherited from the associated-sheaf existence theorem The associated module sheaf exists. An ideal sheaf on is quasi-coherent if, for every affine open , there is an ideal such that is the ideal sheaf associated to the -module .
Depends on
Used by
- Blowup of a scheme along an ideal sheaf Definition
- Embedded deformations of a closed subscheme Definition
- Exceptional subscheme of a blowup Definition
- Fitting ideal sheaves Definition
- Rees algebra sheaf of a finite type ideal Definition
- Square-zero extensions, small extensions and first-order thickenings Definition
- Strict transform of a closed subscheme Definition
- Arithmetic genus, geometric genus and delta invariants Lemma
- Blowups of integral projective complex schemes are projective Lemma
- Blowups restrict to open subschemes of the base Lemma
- Integrality and reducedness of blowups from the Rees charts Lemma
- SNC projective compactifications of smooth quasiprojective varieties Lemma
- Quasi-coherent ideals and closed subschemes Theorem
- Quasi-coherent ideals and closed subschemes, complete route Theorem
- Scheme-theoretic image of a quasi-compact morphism Theorem
- Strict transforms of closed subschemes are blowups of the subscheme Theorem
- The exceptional divisor is the projectivized normal cone Theorem
Dependency tree · two levels
16 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 7 (standard reference, not scraped)