The Stacks project

Exercise 111.35.5. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Define addition of tangent vectors, using Exercise 111.35.4 and a suitable morphism

\[ \mathop{\mathrm{Spec}}(K[\epsilon ]) \longrightarrow \mathop{\mathrm{Spec}}(K[\epsilon _1, \epsilon _2]/(\epsilon _1\epsilon _2)). \]

Similarly, define scalar multiplication of tangent vectors (this is easier). Show that $T_{X/S, x}$ becomes a $\kappa (x)$-vector space with your constructions.

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View Exercise 111.35.5 as pdf