# Unramified morphism

In algebraic geometry, an unramified morphism is a morphism ${\displaystyle f:X\to Y}$ of schemes such that (a) it is locally of finite presentation and (b) for each ${\displaystyle x\in X}$ and ${\displaystyle y=f(x)}$, we have that

1. The residue field ${\displaystyle k(x)}$ is a separable algebraic extension of ${\displaystyle k(y)}$.
2. ${\displaystyle f^{\#}({\mathfrak {m}}_{y}){\mathcal {O}}_{x,X}={\mathfrak {m}}_{x},}$ where ${\displaystyle f^{\#}:{\mathcal {O}}_{y,Y}\to {\mathcal {O}}_{x,X}}$ and ${\displaystyle {\mathfrak {m}}_{y},{\mathfrak {m}}_{x}}$ are maximal ideals of the local rings.

A flat unramified morphism is called an étale morphism. Less strongly, if ${\displaystyle f}$ satisfies the conditions when restricted to sufficiently small neighborhoods of ${\displaystyle x}$ and ${\displaystyle y}$, then ${\displaystyle f}$ is said to be unramified near ${\displaystyle x}$.

Some authors prefer to use weaker conditions, in which case they call a morphism satisfying the above a G-unramified morphism.

## Simple example

Let ${\displaystyle A}$ be a ring and B the ring obtained by adjoining an integral element to A; i.e., ${\displaystyle B=A[t]/(F)}$ for some monic polynomial F. Then ${\displaystyle \operatorname {Spec} (B)\to \operatorname {Spec} (A)}$ is unramified if and only if the polynomial F is separable (i.e., it and its derivative generate the unit ideal of ${\displaystyle A[t]}$).

## Characterization

Given a morphism ${\displaystyle f:X\to Y}$ that is locally of finite presentation, the following are equivalent:[1]

1. f is unramified.
2. The diagonal map ${\displaystyle \delta _{f}:X\to X\times _{Y}X}$ is an open immersion.
3. The relative cotangent sheaf ${\displaystyle \Omega _{X/Y}}$ is zero.