Exercise 111.35.2. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point, let $s = f(x)$. Consider the solid commutative diagram

with the curved arrow being the canonical morphism of $\mathop{\mathrm{Spec}}(\kappa (x))$ into $X$. If $\kappa (x) = \kappa (s)$ show that the set of dotted arrows which make the diagram commute are in one to one correspondence with the set of linear maps

In other words: describe such a bijection. (This works more generally if $\kappa (x) \supset \kappa (s)$ is a separable algebraic extension.)

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