Definition 111.35.3. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. We dub the set of dotted arrows of Exercise 111.35.2 the *tangent space of $X$ over $S$* and we denote it $T_{X/S, x}$. An element of this space is called a *tangent vector* of $X/S$ at $x$.

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