The intrinsic Zariski tangent space

Finite-dimensionality in the locally finite-type case. Choose an affine neighborhood $\mathrm{Spec}B$ of $x$ provided by [F2], with $B$ a finite-type $k$-algebra. By [F3], for some finite list $b_1,\dots ,b_n$ the evaluation map $P=k[x_1,\dots ,x_n]\to B$ is surjective. By [F4] and [F5], $k$ and then $P$ are Noetherian. The preimage in $P$ of any ideal of $B$ is an ideal of $P$; it is finitely generated by [F8], and its generators map to generators of the ideal in $B$. Thus $B$ is Noetherian. If $x$ corresponds to $\mathfrak{p}\subset B$, [F6] identifies ${\mathcal{O}}_{X,x}$ with $B_{\mathfrak{p}}$, and [F7] describes this localization by fractions. For any ideal $J\subseteq B_{\mathfrak{p}}$, its contraction $I=\{b\in B:b\mathrm{/}1\in J\}$ is an ideal of $B$, hence is generated by finitely many $b_1^{\mathrm{\prime }},\dots ,b_r^{\mathrm{\prime }}$ by [F8]. If $b\mathrm{/}s\in J$, then $b\mathrm{/}1=(s\mathrm{/}1)(b\mathrm{/}s)\in J$, so $b\in I$ and $b={\sum }_i{c}_i{b}_i^{\mathrm{\prime }}$; consequently $b\mathrm{/}s={\sum }_i(c_i\mathrm{/}s)(b_i^{\mathrm{\prime }}\mathrm{/}1)$. Conversely each $b_i^{\mathrm{\prime }}\mathrm{/}1$ lies in $J$, so they generate $J$. Thus $B_{\mathfrak{p}}$ is Noetherian. Its maximal ideal ${\mathfrak{m}}_x$ is finitely generated by [F8], say by $a_1,\dots ,a_r$. Modulo ${\mathfrak{m}}_x^2$, every element ${\sum }_i{b}_i{a}_i$ is the $\kappa (x)$-linear combination ${\sum }_i{\bar{b}}_i[a_i]$, so $C_{x}X$ is finite-dimensional by [F1]. The dual is finite-dimensional as well: a surjection $\kappa (x{)}^r\twoheadrightarrow C_{x}X$ induces an injection ${\mathrm{Hom}}_{\kappa (x)}(C_{x}X,\kappa (x))\hookrightarrow {\mathrm{Hom}}_{\kappa (x)}(\kappa (x{)}^r,\kappa (x))$. This uses only finite generating lists and ordinary induction on the polynomial degree; no dependent choice or Axiom of Choice is invoked.

Difference at the generic point of the affine line. Let $X=\mathrm{Spec}k[t]$ and $\eta =(0)$. By [F6] and [F7], its local ring is the field $k(t)$ with maximal ideal zero, so [F1] gives $C_{\eta }X=0$ and hence $T_{\eta }X=0$. On this affine chart [F11] gives ${\mathrm{\Omega }}_{k[t]\mathrm{/}k}=k[t]\mathrm{d}t$; using [F10] and [F12] to pass to the generic stalk gives ${\mathrm{\Omega }}_{X\mathrm{/}k,\eta }=k(t)\mathrm{d}t$. Its residue-field fibre is the one-dimensional $k(t)$-vector space $k(t)\mathrm{d}t$, so [F9] gives $T_{X\mathrm{/}k,\eta }\cong k(t)$. This proves that the intrinsic and relative tangent spaces need not agree at a nonrational point.

Rational-point comparison. If $x$ is $k$-rational, [F13] identifies the intrinsic cotangent space with the relative cotangent space. Taking $k$-linear duals and using [F9] identifies $T_{x}X$ with $T_{X\mathrm{/}k,x}$. No such identification is asserted for nonrational points.