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.

Closed immersions into affine schemes are quotient spectra

Statement

For a ring $A$ , closed immersions $Z \to Spec A$ are, up to unique isomorphism over $Spec A$ , precisely the morphisms $Spec (A / I) \to Spec A$ for ideals $I \subseteq A$ .

Facts & Assumptions

Given: A closed immersion $i : Z \to Spec A$ .

Proof

technique · direct

1.1

By Stacks Project, Tag 01IN, there is an ideal $I \subseteq A$ whose associated quasi-coherent sheaf is the kernel of $O_{Spec A} \to i_{*} O_{Z}$ , and for every distinguished open $D (f)$ one has $i^{- 1} (D (f)) ≅ Spec (A_{f} / I_{f}) .$ In particular, for $f = 1$ this gives $Z ≅ Spec (A / I)$ over $Spec A$ . This avoids the false inference that an arbitrary surjection of sheaves must be surjective on global sections.

given

2.1

Conversely, a quotient $A ↠ A / I$ induces a morphism $Spec (A / I) \to Spec A$ with closed image $V (I)$ ; on every distinguished open it is the quotient $A_{f} ↠ A_{f} / I_{f}$ , so the structure-sheaf map is surjective. Thus it is a closed immersion.

step 1.1

3.1

The kernel ideal and quotient construction recover each other, giving the claimed classification up to unique isomorphism.

step 2.1∎

Depends on

Used by

A closed subset has many scheme structures Counterexample
Two infinitesimal structures at the origin Example
Quasi-coherent ideals and closed subschemes Theorem

Dependency tree · two levels

11 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, Lemma 26.10.1 (standard reference, not scraped)