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.

Fpqc covering morphisms

Definition

Depends on

Dependency tree · 2 levels

Sources