DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-30

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.

Fpqc covering morphisms

Definition

For a scheme morphism $p : S^{'} \to S$ , say that $p$ is flat if for every $x \in S^{'}$ the induced local-ring map $O_{S, p (x)} ⟶ O_{S^{'}, x}$ is flat as a ring map (Flat and faithfully flat modules and ring homomorphisms). Call $p$ an fpqc covering morphism on this page if it is flat, surjective, and quasi-compact (Quasi-compact and quasi-separated morphisms).

This convention concerns one morphism at a time. Under the Stacks Project's family definition of fpqc covers, the singleton family consisting of a flat, surjective, quasi-compact morphism is an fpqc covering. General fpqc covers may be families, and this page does not identify every such family with one quasi-compact morphism.

Depends on

Used by

Faithfully flat scheme morphism Definition
Fpqc covers are universally submersive Lemma
Fpqc descent of properness components Lemma
Properness descends through fpqc base change Theorem

Dependency tree · two levels

8 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 Project, Morphisms of Schemes, §29.26 Definition 29.26.1 (standard reference, not scraped)
Stacks Project, Étale Cohomology, §59.15 Definition 59.15.1 and Example 59.15.3(3) (standard reference, not scraped)
Stacks Project, Topologies on Schemes, §34.9 Definition 34.9.1 (standard reference, not scraped)