Unramified morphism - Misplaced Pages

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

The residue field $k(x)$ is a separable algebraic extension of $k(y)$ .
$f^{\#}({\mathfrak {m}}_{y}){\mathcal {O}}_{x,X}={\mathfrak {m}}_{x},$ 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., $B=A/(F)$ 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 $A$ ).

Curve case

Let $f:X\to Y$ be a finite morphism between smooth connected curves over an algebraically closed field, P a closed point of X and $Q=f(P)$ . We then have the local ring homomorphism $f^{\#}:{\mathcal {O}}_{Q}\to {\mathcal {O}}_{P}$ where $({\mathcal {O}}_{Q},{\mathfrak {m}}_{Q})$ and $({\mathcal {O}}_{P},{\mathfrak {m}}_{P})$ are the local rings at Q and P of Y and X. Since ${\mathcal {O}}_{P}$ is a discrete valuation ring, there is a unique integer $e_{P}>0$ such that $f^{\#}({\mathfrak {m}}_{Q}){\mathcal {O}}_{P}={{\mathfrak {m}}_{P}}^{e_{P}}$ . The integer $e_{P}$ is called the ramification index of $P$ over $Q$ . Since $k(P)=k(Q)$ as the base field is algebraically closed, $f$ is unramified at $P$ (in fact, étale) if and only if $e_{P}=1$ . Otherwise, $f$ is said to be ramified at P and Q is called a branch point.