Unramified morphism explained
of schemes such that (a) it is locally of finite presentation and (b) for each
and
, we have that
- The residue field
is a
separable algebraic extension of
.
f\#(ak{m}y)l{O}x,=ak{m}x,
where
and
are maximal ideals of the local rings.
A flat unramified morphism is called an étale morphism. Less strongly, if
satisfies the conditions when restricted to sufficiently small neighborhoods of
and
, then
is said to be unramified near
.
Some authors prefer to use weaker conditions, in which case they call a morphism satisfying the above a G-unramified morphism.
Simple example
Let
be a ring and
B the ring obtained by adjoining an
integral element to
A; i.e.,
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
).
Curve case
Let
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
where
and
are the local rings at
Q and
P of
Y and
X. Since
is a
discrete valuation ring, there is a unique integer
such that
. The integer
is called the
ramification index of
over
. Since
as the base field is algebraically closed,
is unramified at
(in fact,
étale) if and only if
. Otherwise,
is said to be ramified at
P and
Q is called a
branch point.
Characterization
Given a morphism
that is locally of finite presentation, the following are equivalent:
- f is unramified.
is an open immersion.
is zero.
See also
