ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

A local PID gives a two-point spectrum with one generic point and one closed point

Example

Let $k$ be a field and let $R = k [x]_{(x)}$ . Then $Spec (R) = {(0), (x)},$ with $(0)$ the generic point and $(x)$ the unique closed point.

Facts & Assumptions

Given: A field $k$ and the local PID $R = k [x]_{(x)}$ .

[L1]

Specialisation in a prime spectrum is reverse inclusion (Specialisation in a prime spectrum is reverse inclusion).

[L2]

Closed points of a spectrum are exactly maximal ideals (The closed points of the prime spectrum are exactly the maximal ideals).

[A1]

The only prime ideals of the discrete valuation ring $k [x]_{(x)}$ are $(0)$ and $(x)$ , and $(x)$ is maximal.

Verification

technique · direct

1.1

Fact [A1] gives the full point set $Spec (R) = {(0), (x)}$ . Since $(0) \subseteq (x)$ , fact [L1] shows that $(x)$ is a specialization of $(0)$ , so $(0)$ is the generic point.

L1A1

1.2

By [A1], the ideal $(x)$ is maximal. Therefore [L2] makes $(x)$ the unique closed point of the spectrum.

L2A1

2.1

Thus the specialization poset of $Spec (k [x]_{(x)})$ is the two-point chain $(0) ⇝ (x)$ .

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

J. S. Milne, A Primer of Commutative Algebra, v4.03, §14 (standard reference, not scraped)
The Stacks Project, Section 10.17: The spectrum of a ring (standard reference, not scraped)