CounterexampleConstruction: Literature-sourcedVerification: Literature-sourcedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

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

1.1F1given

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.

2.1F2step 1.1algebra∎

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.

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)