TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Quasi-coherent ideals and closed subschemes

Statement

For a scheme $X$ , the assignments $I \mapsto V (I)$ and $i : Z ↪ X \mapsto \ker (O_{X} \to i_{*} O_{Z})$ give mutually inverse correspondences between quasi-coherent ideal sheaves on $X$ and closed subschemes of $X$ .

Facts & Assumptions

Given: A scheme $X$ .

[F1]

For a ring $A$ , closed immersions into $Spec A$ are precisely quotient-spectrum morphisms $Spec (A / I) \to Spec A$ for ideals $I \subseteq A$ , up to unique isomorphism over $Spec A$ Closed immersions into affine schemes are quotient spectra.

Proof

technique · direct

1.1

Let $I \subseteq O_{X}$ be quasi-coherent and let $U = Spec A$ be an affine open. By its defining local form, $I ∣_{U}$ is associated to an ideal $I_{U} \subseteq A$ , and [F1] gives the closed immersion $Spec (A / I_{U}) \to U$ .

givenF1

1.2

Conversely, let $i : Z ↪ X$ be a closed immersion. Restricting to $U = Spec A$ , [F1] identifies $Z \cap U \to U$ with $Spec (A / I_{U}) \to Spec A$ , and identifies the kernel of $O_{U} \to i_{*} O_{Z \cap U}$ with the ideal sheaf associated to $I_{U}$ . Thus the global kernel is quasi-coherent.

givenF1

2.1

On an overlap of two affine opens, both local quotient immersions have kernel the restricted ideal sheaf $I$ ; hence their quotient rings and their closed immersions agree after restriction. They therefore glue to a closed subscheme, denoted $V (I)$ , of $X$ .

step 1.1

3.1

On every affine open the two constructions are the inverse quotient-ring constructions of [F1]. Since both the ideal sheaves and closed immersions agree on the affine open cover, the constructions are mutually inverse on $X$ .

step 1.1step 2.1step 1.2∎

Depends on

Used by

Scheme-theoretic image of a quasi-compact morphism Theorem

Dependency tree · two levels

8 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, Morphisms of Schemes, Lemma 2.3 (standard reference, not scraped)