LemmaStatement: 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.

Finite type is affine-local on source and target

Statement

Being locally of finite type is affine-local on both source and target. A quasi-compact morphism locally of finite type is of finite type; equivalently, over each affine target open it may be tested on a finite affine source cover.

Facts & Assumptions

Given: A morphism $f : X \to S$ and affine source and target covers.

Proof

technique · direct

1.1

Restricting a finite-type ring map to distinguished affine opens localizes the map and retains a finite generating set, so the condition survives affine refinement.

given

2.1

Conversely, let $V = Spec R$ be an affine target and $U = Spec A \subseteq f^{- 1} (V)$ an affine source open. If a source cover already verifies local finite type, quasi-compactness of $U$ and the principal-open refinement lemma give a finite distinguished cover $U = ⋃_{i = 1}^{n} D (a_{i})$ for which every $A_{a_{i}}$ is a finite-type $R$ -algebra. Choose finitely many localized generators on each member and clear their finitely many denominators. Since $(a_{1}, \dots, a_{n}) = A$ , the standard finite-localization criterion then shows that $A$ is finite type over $R$ . This is the ring argument in Stacks Project, Tag 01T2.

step 1.1algebra

3.1

Refining target overlaps by distinguished opens gives the same argument on the target side. Finally, quasi-compactness of $f$ supplies a finite affine source subcover over each affine target open, so locally finite type plus quasi-compactness is exactly finite type.

step 2.1∎

Depends on

Used by

Affine n-space over an arbitrary base Example

Dependency tree · two levels

16 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 15.2 (standard reference, not scraped)