The Stacks project

Proof. Recall that $\text{Inf}(\mathcal{D}\! \mathit{ef}_ X)$ is the set of automorphisms of the trivial deformation $X' = X \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k[\epsilon ])$ of $X$ to $k[\epsilon ]$ equal to the identity modulo $\epsilon $. By Deformation Theory, Lemma 91.14.2 this is equal to $\text{Ext}^0_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X)$. The equality $\text{Ext}^0_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/k}, \mathcal{O}_ X)$ follows from More on Morphisms of Spaces, Lemma 76.21.4. The equality $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/k}, \mathcal{O}_ X) = \text{Der}_ k(\mathcal{O}_ X, \mathcal{O}_ X)$ follows from More on Morphisms of Spaces, Definition 76.7.2 and Modules on Sites, Definition 18.33.3.

Recall that $T_{x_0}\mathcal{D}\! \mathit{ef}_ X$ is the set of isomorphism classes of flat deformations $X'$ of $X$ to $k[\epsilon ]$, more precisely, the set of isomorphism classes of $\mathcal{D}\! \mathit{ef}_ X(k[\epsilon ])$. Thus the second statement of the lemma follows from Deformation Theory, Lemma 91.14.2. $\square$