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.


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