In the introduction, we have explained the general idea of duality between spaces and rings of functions, and we have given two examples: one linear algebraic and the other topological. We present in what follows an algebro-geometric example. The functions considered will be algebraic. More precisely, the starting
point will be to take as our ring of functions the ring of polynomials
.
Algebraic geometry is concerned with the geometry of spaces whose functions are of this kind. Using different starting rings, it is possible to develop in a parallel way different theories, such as analytic geometry, starting from the ring
of power series converging on an open neighborhood of the origin, or formal geometry, starting from the ring
of all formal power series.
An affine (algebraic) variety over
is the vanishing locus in
of finitely many polynomials
:

Affine varieties define a notion of
space in algebraic geometry.
We take as our notion of ring of functions the finitely generated unital commutative algebras over
. Let
be such an algebra. By hypothesis, there exists a finite number
of generators
of
, i.e.
is the quotient of the polynomial algebra
by some ideal of relations. By the Hilbert basis theorem, i.e. the fact that the ring
is Noetherian, the ideal of relations is itself finitely generated. Let
,...,
be some generators of the relations. We denote by
the ideal generated by the
's, i.e. the set of all elements of the form
,
, i.e. the
-submodule of
generated by the
's.
Then we have an isomorphism:

We associate to such a ring

the affine variety:

For example, we associate

to the ring
![{\displaystyle \mathbb {C} [x_{1},\ldots ,x_{n}]}](//restbase.wikitolearn.org/en.wikitolearn.org/v1/media/math/render/svg/d85aa77dcb275059c2ef9fb836c34c82d14823de)
.
Each point

defines a ring homomorphism:

called the
evaluation map at 
.
The
spectrum of the ring

, denoted by

, is the set of all ring homomorphisms

(since all our rings are in fact

-algebras,
ring homomorphism will always mean
homomorphism of algebras). The evaluation map induces a map

defined by

.
It is possible to show that this map is a bijection, i.e. every ring homorphism

is the evaluation

for some

. In other words, we have

Exercise: Check this is true by assuming that this is known for
,
i.e. that
. The space
seen as an affine variety
is called the affine space of dimension
.
Conversely, starting from an affine variety
, it is possible to construct
a ring of functions on
by quotienting
by the ideal of functions vanishing on
.
The resulting finitely generated
unital algebra is called the coordinate ring of
and is denoted by
.
Example
We have an isomorphism
. The ring
is the coordinate ring of the
-axis in
. The isomorphism
with
corresponds to the obvious geometric fact that the
-axis in
is a copy of
.
Example
We have an isomorphism
given by
. The ring
is the coordinate ring of the parabola
in
.
The isomorphism with
corresponds to the geometric fact that the projection
of the parabola on the
-axis is an isomorphism of affine varieties.
Example
Let
and
. The ring
is the coordinate ring of the graph in
of
for
such that
. The projection on the
-axis gives an isomorphism between this affine variety and the
-axis minus the points where
. In other words, we have isomorphisms
and
.
The ring
is generally denoted by
and is called the localization of
along the multiplicative set
consisting of powers of
.
We have explained how to associate an affine variety to a finitely generated unital
commutative algebra
, by taking its spectrum
, and how to associate a finitely generated unital commutative algebra to an affine variety
, by taking its ring of functions
.
But the constructions
and
are not in general the inverse of each other.
Example
Let
and
.
The equations
and
define the same subset
of
, so we have
, but
.
To solve this difficulty and have a better correspondence between spaces and rings
of functions, there are two possible approaches:
1) Restrict the notion of ring. Rather than to consider rings of the form
for a general ideal
of relations, one can consider only rings of the form
for an ideal which is radical, i.e. such that
where
. With this notion of rings, one obtains a one-to-one correspondence bewteen affine varieties
and rings. This results from the Nullstellensatz which states that
.
2) Enlarge the notion of space. Rather than considering spaces as subsets of
, one
could remember more information. For example, one would like to say that the space associated to the ring
is different from the one associated to the ring
, the former being a first order infinitesimal thickening of the latter. The notion of scheme enlarges the notion of space to take into account the possibility of infinitesimal thickenings. There is a one-to-one correspondence between affine schemes and rings of the form
.
We will describe the first approach in more detail. Given an ideal
we have the associated affine variety
. To an affine variety
we have associated the ideal of polynomials which vanish on it
and have defined its ring of functions to be
. It is straightforward to check that
. But in general
. For example,
. One of the many forms of the Nullstellensatz states that for an affine variety
we have
. So the ring of functions on
is just
, that is
. This gives a bijection between radical ideals and affine varieties which in turn sets up a contravariant equivalence between the category
of affine varieties and the category of finitely-generated reduced algebras (i.e. nilpotent elements, it is easy to see that
is reduced if and only if
is radical).
For instance, given a morphism of varieties
we have an induced homomorphism on the rings of functions
given by
.
Going the other way, given a homomorphism between finitely-generated reduced
-algebras,
, we can choose representations

and define an induced morphism between the associated affine varieties

and

to be the restriction of the polynomial map

,

where

represents

. This gives a well defined morphism when restricted to

. Then, if

and

we have

but

so

and

.
It is straight forward to check that

and

so that isomorphisms in one category induce isomorphisms in the other and vice versa.
Example
Let
,
. The morphism
,
has inverse morphism
. This corresponds to the isomorphisms of rings of functions
,

Example
Let

,

. The morphism

,

induces the following homomorphism on the rings of functions

,

This is clearly not onto and so this is not an isomorphism of varieties. (In fact, there is no surjective homomorphism

. Suppose

and

with

for some

. Then, differentiating,

so

and

are coprime and in particular

and

cannot have any multiple zeros in common. But we also have

so

and

must have the same roots but neither can have any single roots. This is a contradiction.)
There is a table of dualities:

There is an alternative description of
as the set of maximal ideals of
; we will now show that this is equivalent to our original definition. Firstly, if
is a maximal ideal of
then we let
be the quotient morphism
. It is a basic fact (see the accompanying exercises) that
, so we have a morphism
. In the reverse direction, given a ring homomorphism
, we let
. Then
is an ideal of
, and it is maximal because
(since
is an algebra morphism and
), and so
is a field. It is straightforward to check that these constructions are mutually inverse.
Exercise: Show that
, i.e. that for every maximal ideal
of
is of the form
for some (unique)
.
Let
and consider the evaluation map
defined by
. It is clear that the ideal
is contained in the kernel of
. Now, given
we can write it in the form

and see that if

then

Hence,
Conversely, given a maximal ideal
we know by exercise 1 that the quotient is isomorphic to
. If
is identified with
under this isomorphism then
is contained in the natural projection
since
. By maximality, we conclude that it must be equal to
.
Exercise: Using that
, prove the
Nullstellensatz, i.e. for
we have
Checking that
is straightforward.
Let
. Now introduce a new variable
and consider the ideal
. The corresponding variety
must be empty. For if not, there exists some
such that
and yet
since
and
. Now it must be the case that
. Again, supose not, then
lies in some maximal ideal
but then we would have
. So write

for some generators

of

and

. Formally substituting

and then multiplying through by a sufficiently higher power of

we get an expression of the form

and conclude that
Remark:
What we call spectrum is what is generally called the maximal spectrum,
since it is the set of maximal ideals. What is generally called the spectrum is the set
of prime ideals. If we only consider varieties over
, the maximal spectrum is enough.
Exercise: Why the name spectrum? Given
consider the (commutative!)
-algebra generated by
. What is its Spec?