DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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-overlap separation condition

Definition

An $S$ -scheme $X$ satisfies the affine-overlap separation condition if, for every pair of affine opens $U, V \subseteq X$ mapping into a common affine open $W = Spec R \subseteq S$ , the intersection $U \cap V$ is affine and the canonical map $Γ (U, O_{X}) \otimes_{R} Γ (V, O_{X}) ⟶ Γ (U \cap V, O_{X})$ is surjective. This is the affine criterion for usual separatedness; the diagonal formulation is developed later.

Depends on

Used by

Scheme-theoretic varieties Definition
Bertini smoothness away from the base locus Theorem
Irreducible classical varieties and integral separated finite-type schemes Theorem

Dependency tree · two levels

13 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, Schemes, Lemma 21.7 (standard reference, not scraped)