CounterexampleConstruction: Literature-sourcedVerification: Literature-sourcedjudge 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.

Finite type need not mean finite presentation

Statement refuted

Every finite-type morphism is of finite presentation.

Facts & Assumptions

Given: Let $k$ be a field, $A = k [x_{1}, x_{2}, \dots]$ , and $I = (x_{1}, x_{2}, \dots)$ . The ideal $I$ is not finitely generated: a finite list of its elements involves only finitely many variables and cannot generate a later variable.

[F1]

A finite-type morphism is locally defined by finite-type ring maps Locally finite type and finite type morphisms.

[F2]

A locally finite-presentation morphism is locally defined by finitely presented ring maps Locally finite presentation morphisms.

Counterexample

technique · direct

1.1

The quotient map $A \to A / I$ is generated as an $A$ -algebra by the empty set, so it is of finite type and its affine scheme morphism is finite type.

F1given

2.1

The source is the single point corresponding to $I$ . Every affine target neighbourhood contains some $D (f)$ with $f \notin I$ . Modulo $I$ , such an $f$ is a nonzero scalar. The vector space $I / I^{2}$ has basis given by the classes of the $x_{i}$ , and localization at $f$ leaves it infinite-dimensional because $f$ acts on it by that nonzero scalar. Hence $I_{f} / I_{f}^{2} ≅ (I / I^{2})_{f}$ is not finitely generated over $A_{f} / I_{f} ≅ k$ , so $I_{f}$ itself is not finitely generated. Thus $A_{f} \to A_{f} / I_{f}$ is not finitely presented on any target neighbourhood of the source point. By [F2], the affine morphism is not locally of finite presentation and therefore not of finite presentation.

F2step 1.1algebra∎

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, Section 22 (standard reference, not scraped)