TheoremStatement: Literature-sourcedProof: AI-adaptedaudited 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 schemes are quasi-separated

Statement

Every affine scheme is quasi-separated.

Facts & Assumptions

Given: An affine scheme $X$ and affine open subschemes $U, V \subseteq X$ .

Proof

technique · direct

1.1given

Write $X = Spec A$ . Since $U$ and $V$ are quasi-compact open subsets of $X$ , each is a finite union of distinguished opens, say $U = ⋃_{i} D (f_{i})$ and $V = ⋃_{j} D (g_{j})$ .

2.1step 1.1

Hence $U \cap V = ⋃_{i, j} D (f_{i} g_{j})$ is a finite union of quasi-compact distinguished opens, and is therefore quasi-compact.

3.1step 2.1∎

This is exactly the affine-intersection criterion in the definition of quasi-separatedness.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Ravi Vakil, Foundations of Algebraic Geometry, Section 6.1.1 (standard reference, not scraped)