Tangent vectors as dual-number points

Dual-number points are local homomorphisms. Let $\tau \text{\hspace{0.17em}⁣}:D_{\kappa }\to X$ be an $S$-morphism reducing to $x$. Since $D_{\kappa }$ is a one-point scheme with closed point mapping to $x$, [F2] lets us work on an affine chart $U=\mathrm{Spec}B\ni x$ and shows that $\tau$ corresponds to a ring map $B\to \kappa [ϵ]\mathrm{/}({ϵ}^2)$ whose composite with $ϵ\mapsto 0$ is the restriction of the residue map $B\to \kappa$. Passing to the local ring gives a well-defined ring map $\phi \text{\hspace{0.17em}⁣}:R\to \kappa [ϵ]\mathrm{/}({ϵ}^2)$ with $\phi (\mathfrak{m})\subseteq (ϵ)$ and $\phi (a)\equiv a<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>\mathfrak{m}$ for $a\in R$, and the $S$-morphism condition says that $\phi$ restricted to $A={\mathcal{O}}_{S,s}$ lands in $\kappa \subseteq \kappa [ϵ]\mathrm{/}({ϵ}^2)$. Conversely such a $\phi$ determines $\tau$ by the same description on an affine chart containing $x$; two charts give the same morphism by [F2].

Dual-number points are derivations. A ring map $\phi \text{\hspace{0.17em}⁣}:R\to \kappa [ϵ]\mathrm{/}({ϵ}^2)$ with $\phi (a)\equiv a<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>\mathfrak{m}$ has the form $\phi (a)=\bar{a}+ϵD(a)$ with $\bar{a}$ the class of $a$ in $\kappa$ and a unique map $D\text{\hspace{0.17em}⁣}:R\to \kappa$; the map $\phi$ is additive exactly when $D$ is, and $\phi (ab)=\phi (a)\phi (b)$ for all $a,b$ is equivalent to the Leibniz rule $D(ab)=\bar{a}D(b)+\bar{b}D(a)$, since ${ϵ}^2=0$. Moreover $\phi {\mathrm{\mid }}_A$ lands in $\kappa$ exactly when $D$ kills the image of $A$. Hence passage to $D$ is a bijection between the ring maps of step 1.1 and the $A$-derivations $D\text{\hspace{0.17em}⁣}:R\to \kappa$; the derivation is recovered from the product expansion of $\phi$, so the correspondence is natural in $(X,x)$.

Derivations are tangent vectors. Evaluation gives ${\mathrm{Der}}_A(R,\kappa )\cong {\mathrm{Hom}}_R({\mathrm{\Omega }}_{R\mathrm{/}A},\kappa )$ by [F4] (with $N=\kappa$, an $R$-module through $R\to \kappa$), and restriction and extension of scalars along $R\to \kappa$ give ${\mathrm{Hom}}_R({\mathrm{\Omega }}_{R\mathrm{/}A},\kappa )\cong {\mathrm{Hom}}_{\kappa }({\mathrm{\Omega }}_{R\mathrm{/}A}{\otimes }_R\kappa ,\kappa )$. By [F5] the latter is ${\mathrm{Hom}}_{\kappa }({\mathrm{\Omega }}_{X\mathrm{/}S}{\otimes }_{{\mathcal{O}}_{X,x}}\kappa ,\kappa )$, which is the relative tangent space $T_{X\mathrm{/}S,x}$ as recalled in [F1]. Composing the bijections of steps 1.1 and 2.1 with this identification gives the asserted bijection between the dual-number points reducing to $x$ and the relative tangent space.

Naturality and the rational-point case. The correspondence of steps 1.1–3.1 is natural for morphisms of pointed $S$-schemes with a fixed coefficient field $\kappa$: if $h:X\to Y$ sends the chosen $\kappa$-point over $x$ to a $\kappa$-point over $y$, composition sends a map $D_{\kappa }\to X$ to a map $D_{\kappa }\to Y$. On local rings the $ϵ$-coefficient is the derivation $D\circ h_y^{\mathrm{♯}}:{\mathcal{O}}_{Y,y}\to \kappa$, matching pullback of cotangent vectors after tensoring ${\mathrm{\Omega }}_{Y\mathrm{/}S,y}$ with $\kappa$ along $\kappa (y)\to \kappa$. If the residue-field map $\kappa (y)\to \kappa (x)$ is an isomorphism, this is the usual map of relative tangent spaces; for a nontrivial residue-field extension, the target is instead the $\kappa$-dual of the base-extended cotangent space, with no map from $D_{\kappa (x)}$ to $D_{\kappa (y)}$ assumed. In the case $S=\mathrm{Spec}k$ and $x$ $k$-rational, [F6] identifies ${\mathrm{\Omega }}_{X\mathrm{/}k}\otimes \kappa (x)$ with ${\mathfrak{m}}_x\mathrm{/}{\mathfrak{m}}_x^2$, so the bijection becomes the classical tangent-space description. A dual-number point whose $ϵ$-coefficient functional vanishes has $D=0$, hence $\phi$ is the residue map and $\tau$ is the constant morphism, and conversely.