ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-27

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Dual-number vectors in affine space

Example

Let $k$ be a field, $n \geq 0$ , and let $A_{k}^{n} = Spec k [X_{1}, \dots, X_{n}]$ be affine $n$ -space over $k$ with a $k$ -rational point $x = (a_{1}, \dots, a_{n})$ , so that $κ (x) = k$ . Then the $k$ -morphisms from the dual-numbers scheme $D_{k} = Spec k [ϵ] / (ϵ^{2})$ to $X$ that reduce to $x$ are exactly the maps $φ_{v} ⁣ : k [X_{1}, \dots, X_{n}] ⟶ k [ϵ] / (ϵ^{2}), X_{i} ⟼ a_{i} + ϵ v_{i},$ with $v = (v_{1}, \dots, v_{n}) \in k^{n}$ arbitrary and uniquely determined by $φ_{v}$ . Under the bijection of Tangent vectors as dual-number points these are the tangent vectors of $A_{k}^{n}$ at $x$ , and the coefficient $v_{i}$ is the value of the corresponding cotangent functional on the basis element $d X_{i}$ . Thus $T_{A_{k}^{n} / k, x} ≅ k^{n}$ with coordinates $v_{1}, \dots, v_{n}$ .

Facts & Assumptions

Given: A field $k$ , an integer $n \geq 0$ , the polynomial algebra $P = k [X_{1}, \dots, X_{n}]$ , the scheme $X = A_{k}^{n} = Spec P$ over $S = Spec k$ with structure map $k \to P$ , and a $k$ -rational point $x = (a_{1}, \dots, a_{n})$ , whose associated maximal ideal is $m_{x} = (X_{1} - a_{1}, \dots, X_{n} - a_{n})$ with residue field $κ (x) = k$ .

[F1]

Tangent vectors as dual-number points: for a morphism of schemes $f ⁣ : X \to S$ and $x \in X$ with residue field $κ = κ (x)$ , the $S$ -morphisms $D_{κ} \to X$ reducing to the canonical point $x$ are in bijection with ${Hom}_{κ} (Ω_{X / S} \otimes_{O_{X, x}} κ (x), κ (x)) = T_{X / S, x}$ ; for $x$ a $k$ -rational point over $S = Spec k$ this reads ${τ} ≅ {Hom}_{k} (m_{x} / m_{x}^{2}, k)$ .

[F2]

Polynomial differentials are free with $n$ variables: $Ω_{P / k}$ is a free $P$ -module with basis $d X_{1}, \dots, d X_{n}$ ; in particular the $P$ -module is freely generated by the differentials of the coordinates.

[F3]

Universal property of a polynomial ring on an arbitrary family of indeterminates: for commutative rings $R \to S$ and a family $(s_{i})_{i \in I}$ of elements of $S$ , there is exactly one $R$ -algebra homomorphism $R [x_{i} : i \in I] \to S$ sending $x_{i}$ to $s_{i}$ .

[F4]

Affine charts recover the algebraic module of differentials: for the affine morphism $Spec P \to Spec k$ induced by $k \to P$ , the sheaf $Ω_{X / S}$ is the sheaf attached to the $P$ -module $Ω_{P / k}$ , so its fibre at $x$ is $Ω_{P / k} \otimes_{P} κ (x)$ .

Verification

1.1

A $k$ -algebra homomorphism $φ ⁣ : P \to k [ϵ] / (ϵ^{2})$ is the same thing as the data of the $n$ elements $φ (X_{i}) \in k [ϵ] / (ϵ^{2})$ , arbitrary and unique: by [F3] applied to $R = k$ , $S = k [ϵ] / (ϵ^{2})$ , the structure map $k \to k [ϵ] / (ϵ^{2})$ and the family of chosen images. Expanding the unit, a general element of $k [ϵ] / (ϵ^{2})$ is uniquely $c + d ϵ$ with $c, d \in k$ , so $φ$ is uniquely described by the pairs $(c_{i}, d_{i})$ with $φ (X_{i}) = c_{i} + d_{i} ϵ$ .

F3given

1.2

The differential side: by [F2], $Ω_{P / k}$ is free with basis $d X_{1}, \dots, d X_{n}$ , and by [F4] the fibre of $Ω_{X / S}$ at $x$ is $Ω_{P / k} \otimes_{P} κ (x)$ , which after tensoring the basis is the $k$ -vector space with basis the images of $d X_{1}, \dots, d X_{n}$ . Its $k$ -linear dual therefore has the dual basis $\partial_{1}, \dots, \partial_{n}$ with $\partial_{i} (d X_{j}) = δ_{i j}$ , and ${Hom}_{k} (Ω_{X / S} \otimes κ (x), κ (x)) ≅ k^{n}$ by $λ \mapsto (λ (d X_{1}), \dots, λ (d X_{n}))$ .

F2F4given

2.1

Reduction to $x$ : the composite of $φ$ with the quotient map $k [ϵ] / (ϵ^{2}) \to k$ , $ϵ \mapsto 0$ , is a $k$ -algebra homomorphism $P \to k$ ; it is a $k$ -point of $A_{k}^{n}$ and it equals $(a_{1}, \dots, a_{n})$ exactly when $c_{i} = a_{i}$ for all $i$ , by the same uniqueness of [F3] applied to $P \to k$ . Hence the maps $φ$ reducing to $x$ are precisely the $φ$ with $φ (X_{i}) = a_{i} + ϵ d_{i}$ , and they are in bijection with the $n$ -tuples $v = (v_{1}, \dots, v_{n}) = (d_{1}, \dots, d_{n}) \in k^{n}$ .

F3step 1.1given

3.1

By [F1] the dual-number points of step 2.1 are in bijection with the dual space of step 1.2; tracking the $ϵ$ -coefficient, the point $φ_{v}$ corresponds to the functional $λ_{v}$ with $λ_{v} (d X_{i}) = v_{i}$ , that is, $v_{i}$ is the value on the cotangent basis element $d X_{i}$ . Since $x$ is $k$ -rational, this is the identification $T_{A_{k}^{n} / k, x} ≅ {Hom}_{k} (m_{x} / m_{x}^{2}, k) ≅ k^{n}$ .

F1step 1.2step 2.1

4.1

Summing up: the dual-number points of $A_{k}^{n}$ reducing to $x$ are exactly the $φ_{v}$ of step 2.1 with $v \in k^{n}$ , and the bijection of [F1] with the relative tangent space is the one carrying $φ_{v}$ to the functional with coordinates $(v_{1}, \dots, v_{n})$ of step 3.1; in particular the affine space has tangent space $k^{n}$ at each $k$ -rational point, with the coordinate $v_{i}$ dual to $d X_{i}$ .

step 2.1step 3.1F1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Stacks Morphisms 29.33; Vakil 22.2.18 (standard reference, not scraped)