The Stacks project

Definition 26.18.4. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$ be a point. The scheme theoretic fibre $X_ s$ of $f$ over $s$, or simply the fibre of $f$ over $s$, is the scheme fitting in the following fibre product diagram

\[ \xymatrix{ X_ s = \mathop{\mathrm{Spec}}(\kappa (s)) \times _ S X \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(\kappa (s)) \ar[r] & S } \]

We think of the fibre $X_ s$ always as a scheme over $\kappa (s)$.

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View Definition 26.18.4 as pdf