ExampleConstruction: Literature-sourcedVerification: 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.

The scheme-theoretic image of a dense open immersion

Example

Let $X$ be an integral scheme and let $j : U ↪ X$ be a dense open immersion. Then the scheme-theoretic image of $j$ is $X$ itself. This does not assume that $j$ is quasi-compact.

Facts & Assumptions

Given: An integral scheme $X$ and a dense open immersion $j : U ↪ X$ .

[F1]

Every nonempty affine open of an integral scheme is the spectrum of a domain Integral schemes.

[F2]

The scheme-theoretic image, when it exists, is the smallest closed subscheme through which the morphism factors Scheme-theoretic image.

Verification

technique · direct

1.1

Let $Z ↪ X$ be a closed subscheme through which $j$ factors, and let $I$ be its ideal sheaf. On a nonempty affine open $V = Spec A$ , every $a \in I (V)$ restricts to zero on the dense open $U \cap V$ .

given

2.1

Choose a nonempty principal open $D (f) \subseteq U \cap V$ . By [F1], $A$ is a domain and $f \neq 0$ ; the equality $a / 1 = 0$ in $A_{f}$ gives $f^{n} a = 0$ for some $n$ , hence $a = 0$ . Thus $I ∣_{V} = 0$ .

F1step 1.1choose

3.1

The same holds on every nonempty affine open, while the empty opens carry only the zero ideal. Hence $I = 0$ , so every closed factorization of $j$ contains $X$ itself. By [F2], the smallest such factorization is $X$ .

F2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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 29.6.7 (standard reference, not scraped)