Square-zero vector extensions encode tangent vectors with coefficients

For $\varphi :A\to R_W$, compose with $\pi$ and put $a_i=(\pi \varphi )({\bar{t}}_i)$. Every $f\in I(X)$ satisfies $f(a_1,\dots ,a_n)=(\pi \varphi )(\bar{f})=0$, so $a=(a_i)\in V(I(X))=X$ by [F1, F2, F3, F4]; conversely evaluation at each $x\in X(k)$ is a $k$-algebra map $e_x:A\to k$, and the coordinate classes generate $A$, so this identifies ${\mathrm{Hom}}_{k\text{-alg}}(A,k)$ with $X(k)$.

Fix $x=(a_1,\dots ,a_n)$ and put $\mathfrak{m}=\ker e_x$; evaluation is surjective on constants, so $A\mathrm{/}\mathfrak{m}=k$, which makes $\mathfrak{m}$ proper, maximal, and prime. Since $I(X)$ is contained in the polynomial evaluation kernel at $a$ by [F3], and that kernel is generated by $t_1-a_1,\dots ,t_n-a_n$ by telescoping each monomial's factors using [F4], their classes generate $\mathfrak{m}$ and $C_A:=\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$ is finite-dimensional. By [F5, F6, F8, F9, F10, F11], ${\mathcal{O}}_{X,x}=A_{\mathfrak{m}}$ has maximal ideal $\mathfrak{m}{A}_{\mathfrak{m}}$ and localization induces a canonical isomorphism $\theta :C_A\stackrel{\sim }{\to }{C}_{x}X$.

Among maps whose reduction is $e_x$, write uniquely $\varphi (a)=e_x(a)+D(a)$ with $D(a)\in W$; comparing products in $R_W$ shows $D$ is a $k$-derivation for the $A$-module structure on $W$ through $e_x$, with $D(ab)=e_x(a)D(b)+e_x(b)D(a)$, and conversely every such derivation gives a map because $W^2=0$. It kills ${\mathfrak{m}}^2$ and restricts to a linear map $C_A\to W$; conversely, for $h:C_A\to W$, $D(a)=h([a-e_x(a)1])$ defines the inverse derivation, since writing $a=e_x(a)1+u$, $b=e_x(b)1+v$ with $u,v\in \mathfrak{m}$ leaves only the terms $e_x(a)v+e_x(b)u$ modulo ${\mathfrak{m}}^2$.

The canonical tensor-Hom map sends $w\otimes \lambda$ to $[c\mapsto \lambda (c)w]$; by [F14] and the canonical tensor symmetry obtained from [F15], it identifies $W{\otimes }_k{C}_A^{\ast }$ with ${\mathrm{Hom}}_k(C_A,W)$. Dualizing $\theta$ identifies this with $W{\otimes }_k{T}_{x}X$ by [F12, F13], so $\varphi$ maps to $(x,t)$ where $x$ is its reduction and $t$ encodes its induced map $C_A\to W$, and the inverse sends $(x,t)$ to evaluation plus the corresponding derivation from step 1.2. These constructions are canonical and natural in $W$; when $W=k$, $k[ϵ]\mathrm{/}({ϵ}^2)\to R_k$, $ϵ\mapsto (0,1)$, identifies this with [F16].

If $W=0$, then $R_W=k$ and the bijection is $\varphi =e_x\leftrightarrow (x,0)$; if $C_{x}X=0$, then $T_{x}X=0$ and every map over $x$ is evaluation. For the empty algebraic set, outside the variety hypothesis, $k[X]=0$ and no unital map $k[X]\to R_W$ exists, matching the absence of pairs. The construction works for every tensor $t$, including $t=0$, and has no reverse implication. Only the finite coordinate presentation of this fixed $X$ is used to prove finite-dimensionality; the tensor-Hom map is canonical, no family of bases or points is selected, and no Axiom of Choice is used.