Scheme-theoretic fibre ($X_y=X{\times }_Y\mathrm{Spec}\kappa (y)$)

The fibre of a morphism of schemes over $y\in Y$ is its base change to the residue field $\kappa (y)$. For $\mathrm{Spec}B\to \mathrm{Spec}A$, with $y$ corresponding to $\mathfrak{p}$, its fibre ring is $B{\otimes }_A\kappa (\mathfrak{p})$. Its structure sheaf records nilpotents and multiplicities that the set-theoretic inverse image cannot see.

For an injective ring homomorphism between nonzero integral domains $A\to B$, the scheme-theoretic fibre over the generic point of $\mathrm{Spec}A$ is $\mathrm{Spec}(S^{-1}B)$ with $S=A\setminus \{0\}$. Localization at these nonzero elements is a nonzero integral domain, so the generic fibre is a nonempty integral scheme. A surjective morphism between these affine schemes necessarily gives an injective ring map: a prime above $(0)$ contains its kernel.