In algebraic geometry, an unramified morphism is a
morphism 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
.
#
where
and
are maximal ideals of the local rings.
A flat unramified morphism is called an
étale morphism
In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy t ...
. 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 In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' is ...
to ''A''; i.e.,
for some monic polynomial ''F''. Then
is unramified if and only if the polynomial ''F'' is separable (i.e., it and its derivative generate the unit ideal of