ExampleConstruction: Literature-sourcedVerification: Literature-sourcedaudited 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.

The projective line from two affine charts

Example

Let $k$ be a field. Let $U = Spec k [t]$ and $V = Spec k [u]$ . Their principal opens $D (t)$ and $D (u)$ have coordinate rings $k [t, t^{- 1}]$ and $k [u, u^{- 1}]$ . The isomorphism $u \mapsto t^{- 1}$ glues $U$ and $V$ along these opens. With two charts there is no nontrivial triple-overlap cocycle; the resulting scheme is, by definition in this construction, the projective line $P_{k}^{1}$ ; equivalently, the notation here names precisely this two-chart gluing.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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 5.5.4 (standard reference, not scraped)