- Source: Unramified morphism
In algebraic geometry, an unramified morphism is a morphism
f
:
X
→
Y
{\displaystyle f:X\to Y}
of schemes such that (a) it is locally of finite presentation and (b) for each
x
∈
X
{\displaystyle x\in X}
and
y
=
f
(
x
)
{\displaystyle y=f(x)}
, we have that
The residue field
k
(
x
)
{\displaystyle k(x)}
is a separable algebraic extension of
k
(
y
)
{\displaystyle k(y)}
.
f
#
(
m
y
)
O
x
,
X
=
m
x
,
{\displaystyle f^{\#}({\mathfrak {m}}_{y}){\mathcal {O}}_{x,X}={\mathfrak {m}}_{x},}
where
f
#
:
O
y
,
Y
→
O
x
,
X
{\displaystyle f^{\#}:{\mathcal {O}}_{y,Y}\to {\mathcal {O}}_{x,X}}
and
m
y
,
m
x
{\displaystyle {\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
{\displaystyle f}
satisfies the conditions when restricted to sufficiently small neighborhoods of
x
{\displaystyle x}
and
y
{\displaystyle y}
, then
f
{\displaystyle f}
is said to be unramified near
x
{\displaystyle 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
{\displaystyle A}
be a ring and B the ring obtained by adjoining an integral element to A; i.e.,
B
=
A
[
t
]
/
(
F
)
{\displaystyle B=A[t]/(F)}
for some monic polynomial F. Then
Spec
(
B
)
→
Spec
(
A
)
{\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
A
[
t
]
{\displaystyle A[t]}
).
Curve case
Let
f
:
X
→
Y
{\displaystyle 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
)
{\displaystyle Q=f(P)}
. We then have the local ring homomorphism
f
#
:
O
Q
→
O
P
{\displaystyle f^{\#}:{\mathcal {O}}_{Q}\to {\mathcal {O}}_{P}}
where
(
O
Q
,
m
Q
)
{\displaystyle ({\mathcal {O}}_{Q},{\mathfrak {m}}_{Q})}
and
(
O
P
,
m
P
)
{\displaystyle ({\mathcal {O}}_{P},{\mathfrak {m}}_{P})}
are the local rings at Q and P of Y and X. Since
O
P
{\displaystyle {\mathcal {O}}_{P}}
is a discrete valuation ring, there is a unique integer
e
P
>
0
{\displaystyle e_{P}>0}
such that
f
#
(
m
Q
)
O
P
=
m
P
e
P
{\displaystyle f^{\#}({\mathfrak {m}}_{Q}){\mathcal {O}}_{P}={{\mathfrak {m}}_{P}}^{e_{P}}}
. The integer
e
P
{\displaystyle e_{P}}
is called the ramification index of
P
{\displaystyle P}
over
Q
{\displaystyle Q}
. Since
k
(
P
)
=
k
(
Q
)
{\displaystyle k(P)=k(Q)}
as the base field is algebraically closed,
f
{\displaystyle f}
is unramified at
P
{\displaystyle P}
(in fact, étale) if and only if
e
P
=
1
{\displaystyle e_{P}=1}
. Otherwise,
f
{\displaystyle f}
is said to be ramified at P and Q is called a branch point.
Characterization
Given a morphism
f
:
X
→
Y
{\displaystyle f:X\to Y}
that is locally of finite presentation, the following are equivalent:
f is unramified.
The diagonal map
δ
f
:
X
→
X
×
Y
X
{\displaystyle \delta _{f}:X\to X\times _{Y}X}
is an open immersion.
The relative cotangent sheaf
Ω
X
/
Y
{\displaystyle \Omega _{X/Y}}
is zero.
See also
Finite extensions of local fields
Ramification (mathematics)
References
Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32. doi:10.1007/bf02732123. MR 0238860.
Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Kata Kunci Pencarian:
- Unramified morphism
- Diagonal morphism (algebraic geometry)
- Étale morphism
- Ramification (mathematics)
- Finite extensions of local fields
- Smooth morphism
- Frobenius endomorphism
- Flat morphism
- Morphism of schemes
- Formally étale morphism
