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
\[ \xymatrix{ \mathop{\mathrm{Spec}}(\kappa (x)) \ar[r] \ar[dr] \ar@/^1pc/[rr] & \mathop{\mathrm{Spec}}(\kappa (x)[\epsilon ]) \ar@{.>}[r] \ar[d]& X \ar[d] \\ & \mathop{\mathrm{Spec}}(\kappa (s)) \ar[r] & S } \]
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
\[ \mathop{\mathrm{Hom}}\nolimits _{\kappa (x)}( \frac{\mathfrak m_ x}{\mathfrak m_ x^2 + \mathfrak m_ s\mathcal{O}_{X, x}}, \kappa (x)) \]
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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