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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