DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Affine morphisms

Definition

A morphism of schemes $f : X \to S$ is affine when $f^{- 1} (U)$ is affine for every affine open subscheme $U \subseteq S$ . Here the inverse image carries the restricted structure sheaf, as in Affine open subschemes, and $f$ is a morphism of locally ringed spaces as in Morphisms of schemes. The empty scheme is affine, being $Spec 0$ . Affineness of a morphism does not require its total source or target to be affine.

Depends on

Used by

Affine schemes and affine morphisms are separated Corollary
Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
Finite morphisms of schemes Definition
Geometric vector bundle with the sections convention Definition
The cotangent complex of a morphism of schemes Definition
Affine morphisms are separated Lemma
Affine pushforward algebra localizes Lemma
Affine pushforward is compatible with sheaf cohomology Lemma
Affineness is local on the target Lemma
Base change and composition of affine morphisms Lemma
Closed immersion preserves cohomology and coherent pushforward Lemma
Composite of a finite morphism and a proper morphism is proper Lemma
Finite is affine and local on its target Lemma
Affine morphisms are relative spectra Theorem
Higher direct images of quasi-coherent modules vanish along affine morphisms Theorem

Dependency tree · two levels

5 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

Stacks 29.11.1 (standard reference, not scraped)