In algebraic geometry, an unramified morphism is a morphism
of schemes such that (a) it is locally of finite presentation and (b) for each
, we have that A flat unramified morphism is called an étale morphism.
Less strongly, if
satisfies the conditions when restricted to sufficiently small neighborhoods of
Some authors prefer to use weaker conditions, in which case they call a morphism satisfying the above a G-unramified morphism.
be a ring and B the ring obtained by adjoining an integral element to A; i.e.,
for some monic polynomial F. Then
Spec (
) → Spec (
{\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
be a finite morphism between smooth connected curves over an algebraically closed field, P a closed point of X and
We then have the local ring homomorphism
are the local rings at Q and P of Y and X.
is a discrete valuation ring, there is a unique integer
The integer
is called the ramification index of
as the base field is algebraically closed,
(in fact, étale) if and only if
is said to be ramified at P and Q is called a branch point.
Given a morphism
that is locally of finite presentation, the following are equivalent:[2]
This algebraic geometry–related article is a stub.
You can help Wikipedia by expanding it.