DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Ideal sheaves

Definition

An ideal sheaf on a scheme $X$ is a subsheaf $I \subseteq O_{X}$ such that $I (U)$ is an ideal of $O_{X} (U)$ for every open $U$ , compatibly with restriction.

Depends on

Schemes

Used by

Closed immersions of schemes Definition
Formally smooth morphism Definition
Formally unramified morphism Definition
Limits of one-parameter orbits and concentrator subschemes Definition
Order of an ideal sheaf at a point Definition
Quasi-coherent ideal sheaves Definition
The coefficient ideal of a marked ideal of maximal order Definition
The homogenized ideal of a marked ideal of maximal order Definition
The monomial part, the non-monomial part and the companion ideal Definition
The reduction of a scheme Definition
Addition and multiplication of marked ideals Lemma
Controlled transforms are well defined Lemma
Schematic closure and agreement on a dense open Lemma
Adjunction for smooth plane curves Theorem
Conormal sequence for a closed immersion Theorem
Effective Cartier divisors are closed subschemes cut out by regular equations Theorem
Serre global-generation criterion for ampleness Theorem
Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme Theorem

Dependency tree · two levels

3 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 8 (standard reference, not scraped)