# Unramified morphism

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

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

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

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

## Simple example

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

## Characterization

Given a morphism $f:X\to Y$ that is locally of finite presentation, the following are equivalent:

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

## See also

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