THE CATEGORY OF AFFINE
k
VARIETIES
PETE L. CLARK
1.
The category of affine varieties over a field
k
Let
k
be an arbitrary field. We wish to define a
category
of affine varieties over
k
.
Recall the most classical definition:
if
k
=
k
, then an affine subvariety of
k
n
is
given by the zero locus of a set (which, by the Hilbert basis theorem, can be taken
to be finite) of polynomials
P
1
(
x
)
, . . . , P
r
(
x
), where
x
= (
x
1
, . . . , x
n
).
Even in the classical case, this is a somewhat creaky definition since it is so ex
trinsic: it is like defining a complex manifold as a certain type of subset of
C
n
. But
in the case of a general field, it is manifestly inadequate: we would like to regard
the equation
x
2
+
y
2
=

1 as defining an affine curve in
R
2
, but this curve has no
(real) points.
There are several rather elaborate remedies for this: we could switch to the theory
of schemes, or we could introduce the theory of Galois descent.
But let us try
something more modest: we switch our arrows around. Namely, let’s try to define
a category Aff
op
k
, the opposite of the category of affine varieties.
In this category, the objects will be
affine kalgebras
, i.e., finitely generated
k

algebras
A
. In other words,
A
is a ring, equipped with a morphism of rings
k
→
A
,
which is isomorphic, as a
k
algebra to
k
[
x
1
, . . . , x
n
]
/I
for some
n
∈
Z
+
. Of course
there will be many such isomorphisms, and we do not privilege any one of them as
part of the definition. A morphism of affine algebras is just what is sounds like: a
morphism of rings which respects the
k
module structure.
Then an
affine variety
V
k
will be identified with a corresponding affine alge
bra
A
V
. However, a morphism
V
→
W
of affine varieties is precisely a morphism
W
k
→
V
k
, i.e., a morphism of the corresponding affine algebras, but in the other
direction.
Exercise:
Show that the category of affine
k
algebras is closed under quotients,
finite Cartesian products, and finite tensor products (over
k
), but is not closed
under passage to arbitrary subalgebras.
Before we go any farther, let us introduce the following construction: let
k
→
k
be
any field extension. Then we have a functor from the category of affine
k
algebras
to the category of affine
k
algebras: we map
A
to
A
k
:=
A
⊗
k
k
. The only thing we
need to check is that if
A
is finitely generated over
k
, then
A
k
is finitely generated
over
k
, but this is immediate: we can write
A
=
k
[
x
]
/I
and then
A
k
∼
=
k
[
x
]
/I
.
1
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PETE L. CLARK
We refer to this process as
base extension
.
Now we can use commutative algebra to define “geometric properties” of affine
varieties. For instance, recall that a commutative ring
A
is called
reduced
if it has
no nonzero nilpotent elements: equivalently, its
nilradical
rad(
A
) – the intersec
tion of all prime ideals – is equal to 0. If we have an affine algebra which is not
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 Fall '10
 Clark
 Algebra, Algebraic geometry, pete l. clark, aﬃne, aﬃne variety

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