ExampleConstruction: Literature-sourcedVerification: AI-adaptedverified 2026-09-26 (gpt-6-sol)

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.

Dual numbers give a one-point nonreduced affine scheme

Example

Let $k$ be a field and $R = k [ϵ] / (ϵ^{2})$ . Then $Spec R$ has one point $(ϵ)$ , its residue field is $k$ , and it is not reduced.

Facts & Assumptions

Given: A field $k$ and $R = k [ϵ] / (ϵ^{2})$ .

[F1]

Passing from a ring to its quotient by the nilradical does not change the underlying prime spectrum (Passing to the reduced quotient does not change the prime spectrum).

Verification

technique · direct

1.1F1givenalgebra

Every element of $R$ has the form $a + b ϵ$ . If $a \neq 0$ , then $a + b ϵ$ is a unit with inverse $a^{- 1} - a^{- 2} b ϵ$ ; while $(b ϵ)^{2} = 0$ . Thus $Nil (R) = (ϵ)$ , and $R / (ϵ) ≅ k$ . Since the field $k$ has exactly one prime ideal, $(0)$ , [F1] identifies $Spec R$ with its one-point spectrum; its sole prime is $(ϵ)$ .

1.2given

The nonzero class of $ϵ$ squares to zero, so $R$ is not reduced.

2.1step 1.1

Every element outside $(ϵ)$ is a unit by step 1.1, so $R_{(ϵ)} = R$ . The residue-field definition therefore gives $κ ((ϵ)) = R_{(ϵ)} / (ϵ) R_{(ϵ)} ≅ R / (ϵ) ≅ k$ .

3.1step 2.1step 1.2∎

This is the asserted one-point nonreduced scheme.

Depends on

Used by

Dense regular loci on every component 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

James S. Milne, Algebraic Geometry, 10.28 (standard reference, not scraped)