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The Stacks project

Lemma 33.16.4. Let f : X \to S be a morphism of schemes. Let x \in X. There is a canonical isomorphism

T_{X/S, x} = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X, x}}(\Omega _{X/S, x}, \kappa (x))

of vector spaces over \kappa (x).

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

\theta ^*\Omega _{X/S} \to \Omega _{\mathop{\mathrm{Spec}}(\kappa [\epsilon ])/\mathop{\mathrm{Spec}}(\kappa (s))} \to \Omega _{\mathop{\mathrm{Spec}}(\kappa [\epsilon ])/\mathop{\mathrm{Spec}}(\kappa )}

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


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