The Stacks project

Proof. Set $\kappa = \kappa (x)$. Given $\theta \in T_{X/S, x}$ we obtain a map

Taking sections we obtain an $\mathcal{O}_{X, x}$-linear map $\xi _\theta : \Omega _{X/S, x} \to \kappa \text{d}\epsilon $, i.e., an element of the right hand side of the formula of the lemma. To show that $\theta \mapsto \xi _\theta $ is an isomorphism we can replace $S$ by $s$ and $X$ by the scheme theoretic fibre $X_ s$. Indeed, both sides of the formula only depend on the scheme theoretic fibre; this is clear for $T_{X/S, x}$ and for the RHS see Morphisms, Lemma 29.32.10. We may also replace $X$ by the spectrum of $\mathcal{O}_{X, x}$ as this does not change $T_{X/S, x}$ (Schemes, Lemma 26.13.1) nor $\Omega _{X/S, x}$ (Modules, Lemma 17.28.7).

Let $(A, \mathfrak m, \kappa )$ be a local ring over a field $k$. To finish the proof we have to show that any $A$-linear map $\xi : \Omega _{A/k} \to \kappa $ comes from a unique $k$-algebra map $\varphi : A \to \kappa [\epsilon ]$ agreeing with the canonical map $c : A \to \kappa $ modulo $\epsilon $. Write $\varphi (a) = c(a) + D(a) \epsilon $ the reader sees that $a \mapsto D(a)$ is a $k$-derivation. Using the universal property of $\Omega _{A/k}$ we see that each $D$ corresponds to a unique $\xi $ and vice versa. This finishes the proof. $\square$