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.

Scheme-theoretic image of a quasi-compact morphism

Statement

Let $f : X \to Y$ be a quasi-compact morphism of schemes and put $I = \ker (O_{Y} \to f_{*} O_{X})$ . Then $I$ is a quasi-coherent ideal sheaf, and the closed subscheme $V (I)$ is the scheme-theoretic image of $f$ . For every open $W \subseteq Y$ , its restriction $V (I) \cap W$ is the scheme-theoretic image of $f^{- 1} (W) \to W$ .

Facts & Assumptions

Given: A quasi-compact morphism $f : X \to Y$ .

[F1]

A morphism is quasi-compact when inverse images of quasi-compact opens are quasi-compact Quasi-compact and quasi-separated morphisms.

[F2]

For a scheme, quasi-coherent ideal sheaves and closed subschemes are in mutually inverse correspondence Quasi-coherent ideals and closed subschemes.

Proof

technique · direct

1.1

Let $V = Spec A$ be an affine open of $Y$ . By [F1], $f^{- 1} (V)$ is quasi-compact, so its affine-open cover has a finite subcover $U_{1}, \dots, U_{n}$ ; when $f^{- 1} (V)$ is empty, take the finite empty cover.

givenF1choose

2.1

Write $U_{i} = Spec B_{i}$ . The sheaf condition makes $Γ (f^{- 1} (V), O_{X})$ inject into $\prod_{i} B_{i}$ , so the kernel of $A \to Γ (f^{- 1} (V), O_{X})$ equals the kernel $I_{V}$ of $A \to \prod_{i} B_{i}$ . The latter is an ideal of $A$ , and its localizations give the corresponding kernels on principal opens of $V$ .

step 1.1

3.1

Thus $I ∣_{V}$ is the ideal sheaf associated to $I_{V}$ on every affine $V$ , so $I$ is quasi-coherent.

step 2.1

4.1

By [F2], $I$ defines a closed subscheme $V (I)$ . On each affine $V$ , the map $A \to \prod_{i} B_{i}$ factors through $A / I_{V}$ , so $f$ factors through it. If a closed subscheme defined on $V$ by $J \subseteq A$ also receives $f$ , then $J \subseteq I_{V}$ ; hence $V (I_{V})$ factors through that competing closed subscheme. This proves minimality.

F2step 3.1

5.1

Repeating the kernel computation on an open $W \subseteq Y$ gives the restriction of $I$ to $W$ . The correspondence in [F2] therefore identifies $V (I) \cap W$ with the scheme-theoretic image of $f^{- 1} (W) \to W$ .

F2step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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 29.6.3 (standard reference, not scraped)