The Stacks project

Exercise 111.35.10. Let $f : X \to Y$ be a morphism of schemes over $S$. Let $x \in X$ be a point. Set $y = f(x)$. Assume that the natural map $\kappa (y) \to \kappa (x)$ is bijective. Show, using the definition, that $f$ induces a natural linear map

\[ \text{d}f : T_{X/S, x} \longrightarrow T_{Y/S, y}. \]

Match it with what happens on local rings via Exercise 111.35.2 in case $\kappa (x) = \kappa (s)$.

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View Exercise 111.35.10 as pdf