# Morphisms in Algebraic Geometry

Next we study the “nice” functions between closed subspaces of .

Definition.Suppose and are closed subsets. A

morphismis a function which can be expressed as:for some polynomials . We also say f is a

regular map.

**Example**

- Let be any closed subset. Regular maps of the form are given by polynomials .
- Take and . Define by . We write this as .
- Take and . Define by . We write this as .

[Example 3: image edited from GeoGebra plot.]

The first example, although basic, is of huge importance.

Definition.A

regular functionfor a closed subset is a morphism of the form .The set of such functions is called the

coordinate ringof V and is denoted by .

**Note**

Each regular function is given by and two polynomials *f*, *g* induce the same function on *V* if and only if

Thus we have , which gives *k*[*V*] its ring structure. To describe this ring structure in terms of regular functions, we have:

# Morphisms in Algebra

General morphisms can now be described in the language of their coordinate rings.

Proposition 1.Let and be closed. There is a bijection between:

- morphisms ;
- ring homomorphisms which are linear over k.

**Note**

The second condition can be rephrased as: is a homomorphism of *k*-algebras. In a later article, we will cover algebras over a ring in greater detail.

**Proof**

Given a morphism , upon composing with a regular function , we obtain a regular function of *V*. This gives a ring homomorphism which is clearly linear over *k*.

Conversely, suppose is a ring homomorphism which is linear over *k*. Let be the images of in . Let which is a regular function . We claim that the function

has image in *W*. Indeed for any we have

which is 0 since in *k*[*W*]. This creates a bijection. ♦

**Note**

In order to swap we require to be a ring homomorphism linear over *k*. E.g. if with then

The following properties are obvious:

Lemma.

- For any closed set V, we have .
- For any morphisms of closed sets we have

**Proof**

The first is clear. For the second, pick any , a regular map . Then

. ♦

## Examples

Let us interpret the earlier examples as homomorphisms .

**Example 2.** We have:

.

We wrote this as for a reason, for *f* corresponds to:

**Example 3.** Similarly

was written as . Algebraically,

# Properties

Since we defined a topology on closed sets, the morphisms should be continuous.

Proposition 2.A morphism is a continuous map, with respect to the subspace topology on both sets.

**Proof**

Let *W’* be a closed subset of *W*, so it is also a closed subset of . We need to show is closed in *V* (equivalently, in . Now *W’* can be written as

for some subset . It follows that

is cut out from *V* by equations . Hence is closed. ♦

## Isomorphisms

Definition.Closed subsets and are said to be

isomorphicif there exist regular maps and such that and .

This implies:

Hence isomorphism of the closed subsets corresponds to isomorphism of the underlying *k*-algebras!

**Example**

Let us take example 3 from above, where is defined by . Note that *f* is bijective on the points, and it corresponds to:

The image of is not surjective since it does not contain *T*. We have thus learnt:

There exist bijective regular maps which are not isomorphisms.

# More General Correspondence

Putting it together, we obtain the following bijective correspondences:

- The top correspondence was the original one.
- The left correspondence follows from point-set topology.
- The right correspondence follows from the correspondence between ideals of and ideals of containing .
- The correspondence preserves radical ideals because is a radical ideal of
*A*if and only if is a reduced ring; now apply .

- The correspondence preserves radical ideals because is a radical ideal of

Summary.In other words, we have a bijection between radical ideals of the coordinate ring k[V] and closed subsets of V. This enables us to look at V and its coordinate ring k[V], ignoring the ambient affine space it sits in.

Could you please explain further the last sentence “This enables us to look at V and its coordinate ring k[V], ignoring the ambient affine space it sits in.”? Is it not the case that we need to consider some ambient affine space while we consider some closed subspace V it sits in?

Yes, for now. In a few articles, you will see how this pan out, where we define an abstract variety as a k-algebra.