---
tag: 963QWX2E2Z
kind: concept
name: Fiber of a morphism
msc: [14A15]
sources:
  - url: https://en.wikipedia.org/wiki/Fiber_product_of_schemes
---

## Summary

The fiber of a [[morphism of schemes|JXRKH6P1Z4]] $f \colon X \to Y$ over a point $y$ is a scheme $X_y$ over the [[residue field|G51E8MW7J1]] $\kappa(y)$.
Its underlying space is the preimage $f^{-1}(y)$, but it also carries a scheme structure, which may be non-reduced.
A morphism is often thought of as a family of schemes, its fibers, parametrized by the points of $Y$.

## Explanations

### L1 human

The fiber over $y$ is "everything in $X$ that lies above $y$", made into a scheme of its own.
Take $\operatorname{Spec} \Z[i] \to \operatorname{Spec} \Z$.
Over the prime $5$ the fiber has two points, because $x^2 + 1$ factors modulo $5$.
Over the prime $3$ it has one point with residue field the field of $9$ elements.
Over the prime $2$ it has one point, but the fiber is $\operatorname{Spec} \F_2[x]/((x+1)^2)$, which has a nonzero nilpotent: this records that $2$ ramifies.