---
tag: 1JQX6HD8E5
kind: concept
name: Unramified morphism
msc: [14A15, "14E22"]
sources:
  - url: https://en.wikipedia.org/wiki/Unramified_morphism
same_as:
  wikidata: Q85812355
  wikipedia: Unramified morphism
---

## Summary

An unramified morphism of schemes is a morphism, locally of finite type, whose fibres are disjoint unions of spectra of finite separable field extensions.
It extends to schemes the notion of an unramified extension of number fields.
An unramified morphism that is also flat and locally of finite presentation is étale.

## Explanations

### L1 human

A map is unramified when no two sheets of it come together at a point.
For the squaring map $t \mapsto t^2$ on the line, over a field of characteristic not $2$, each nonzero value has two preimages.
The value $0$ has only one preimage, where the two sheets touch: the map is ramified there and unramified elsewhere.
In number theory, a prime $p$ ramifies in a number field when $p$ has a repeated prime ideal factor in the ring of integers.
This is the same phenomenon.