Fiber of a morphism

Remarks

The projection $X_y \to X$ is a homeomorphism of the underlying space of $X_y$ onto $f^{-1}(y)$. Example: let $f \colon \operatorname{Spec} \Z[i] \to \operatorname{Spec} \Z$ and $y = (p)$ for a prime $p$. Then $X_y = \operatorname{Spec}(\Z[i] \otimes_\Z \F_p) = \operatorname{Spec} \F_p[x]/(x^2 + 1)$. For $p = 5$ this is $\operatorname{Spec} \F_5[x]/((x-2)(x+2))$, two points. For $y = (0)$ the fiber is $\operatorname{Spec} \Q(i)$, the generic fiber. Non-example: the fiber is in general not the reduced point structure on $f^{-1}(y)$. Let $k$ be a field with $2 \ne 0$, let $f \colon \operatorname{Spec} k[t] \to \operatorname{Spec} k[x]$ be given by $x \mapsto t^2$, and let $y = (x)$. Then $X_y = \operatorname{Spec} k[t]/(t^2)$, which is not reduced, while the single point of $f^{-1}(y)$ with its reduced structure is $\operatorname{Spec} k$.