DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-06

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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.

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The affine scheme of dual numbers

Definition

For a field $k$ , the dual-numbers scheme is $D_{k} = Spec (k [ϵ] / (ϵ^{2}))$ . Its class $ϵ$ is nilpotent, so this is an infinitesimal affine test scheme rather than a reduced point.

Depends on

Used by

A locally principal subscheme need not be an effective Cartier divisor Counterexample
A scheme is not determined by its underlying topological space Counterexample
Smoothness over a field by geometric regularity Definition
The intrinsic Zariski tangent space Definition
The Lie algebra of a group scheme Definition
The scheme-theoretic tangent cone at a point Definition
Classical bilinear-form equations linearize to matrix spaces Example
Differentials of dual numbers in both characteristics Example
Dual numbers compute the tangent spaces of the general and special linear groups Example
Dual numbers give a one-point nonreduced affine scheme Example
A tangent direction is realized by a local smooth curve Lemma
A transverse hyperplane slice is smooth at the chosen point Lemma
Differential of an S-morphism Lemma
Tangent spaces of products over a field Lemma
Tangent vectors at rational points are dual-number points Lemma
The scheme-theoretic linear span of the tangent cone Lemma
The tangent space at the identity is a vector space, and Lie is a functor Lemma
Tangent dimension bounds local dimension Theorem
Tangent vectors as dual-number points Theorem
The Jacobian kernel computes the tangent space Theorem

Dependency tree · two levels

12 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, 10.29 (standard reference, not scraped)