Definition 33.16.3. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. The set of dotted arrows making (33.16.1.1) commute with its canonical $\kappa (x)$-vector space structure is called the tangent space of $X$ over $S$ at $x$ 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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