FIBER PRODUCTS; SEPARATED AND PROPER MORPHISMS
PETE L. CLARK
1.
Fiber products and base change
A very important tool across geometry is the fibered product. Here the basic state
ment is simple: given two
S
schemes
X
and
Y
, the fiber product
X
×
S
Y
exists in
the category of schemes. Namely, this is an
S
scheme, endowed with
S
morphisms
π
X
,
π
Y
to
X
and
Y
, satisfying the following universal mapping property: given
any
S
scheme
Z
and
S
morphisms
f
:
Z
→
X
,
g
:
Z
→
Y
, there exists a unique
morphism
f
×
S
g
:
Z
→
X
×
S
Y
making the usual diagram (see e.g. Liu, p. 79)
commutative.
As usual, the first thing to do is consider the affine case with all the arrows re
versed. In this case, the object which makes the diagram commute is nothing other
than the tensor product
R
⊗
A
S
. The general case is, as usual, reduced to the affine
case by glueing (and, as is often the case, this requires some nontrivial work).
Most often we take fiber products over an affine scheme Spec
A
, and we will abuse
notation by writing
X
×
A
Y
rather than
X
×
Spec
A
Y
. In particular, for any field
(or indeed any ring)
k
we have
A
m
k
×
k
A
n
k
∼
=
A
m
+
n
k
,
this being the geometric analogue of the usual statement about tensor products of
polynomial rings.
A useful fact, which follows immediately from the universal property, is:
(
X
×
S
Y
)(
S
) =
X
(
S
)
×
Y
(
S
)
.
If
X
is an
S
scheme and
S
is another
S
scheme, then we can form the fiber prod
uct
X
×
S
S
. This is still an
S
scheme, but it is also an
S
scheme via the second
projection map. This process – preciesly, of starting with a scheme over
S
taking
the fiber product with a second scheme
S
over
S
, and regarding the fiber product
as an
S
scheme, is called
base change
. We denote the base change by
X
/S
. In
the case of a field extension
k
→
l
and a
k
scheme
X
, the base change
X
/l
is a
scheme over
l
. When
X
is an affine
k
variety, this recovers our previous process of
“base extension” by tensorization from
k
to
l
. Note that in general a base change
from a Spec
R
scheme to a Spec
S
involves a homomorphism
R
→
S
which need
not be injective, hence we use the term “base change” rather than “base extension.”
An extremely important application of base change is to the notion of
fibers
of a
morphism.
Let
f
:
X
→
Y
be a morphism of schemes, and
y
∈
Y
be any point
(closed or otherwise). We want to define a scheme
X
y
, the
fiber of f over y
in a
1
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PETE L. CLARK
way which generalizes the usual idea of fibers of a map of sets. The key to this is
to use the canonical morphism Spec
k
(
y
)
→
Y
. Then we can define
X
y
:=
X
×
Y
Spec
k
(
y
)
,
which is a scheme over the field Spec
k
(
y
).
So for instance, if
Y
is any scheme and
y
∈
Y
, the fiber of
P
n
Y
over
y
is the
usual projective space over the residue field
k
(
y
).
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 Fall '10
 Clark
 Algebra, Geometry, Topology, Algebraic geometry, Algebraic variety, base change

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