# Dimension of relative tangent space

Let $$k$$ be an arbitrary field, $$X$$ be a $$k$$-Scheme locally of finite type, $$x \in X$$ a closed point and $$\kappa(x)$$ its residue field.

Question: Is the dimension of of the tangent space $$T_x X$$ equal to that of the relative tangent space $$T_x (X/k)$$ for the $$\kappa(x)$$-point corresponding to $$x$$? (Both refer to the dimension over the field $$\kappa(x)$$)

I think this question immediately reduces to the affine case where $$\DeclareMathOperator{\Spec}{Spec} X = \Spec A$$ for some finitely generated $$k$$-algebra $$A$$ and $$x$$ corresponds to a maximal ideal $$\newcommand{\m}{\mathfrak{m}} \m \subset A$$. Then

$$\dim_{\kappa(x)} T_x X = \dim_{\kappa(x)} \m/\m^2$$ and $$\dim_{\kappa(x)} T_x (X/k) = \dim_{\kappa(x)} (\m \otimes_k \kappa(x))/(\m \otimes_k \kappa(x))^2$$

By looking at the number of generators of $$\m$$ I think it follows that $$\dim T_x X \leq T_x(X/k)$$ but I'm not sure about the other direction.