affine algebraic variety

Definition

An affine algebraic variety is (informally and loosely) a very special collection of solutions to polynomials. Generally, we want to study these geometrically and algebraically to understand the polynomial or the set of solutions.

Best, most abstract (and involved) definition

An affine algebraic variety is an affine scheme where the ring is finitely generated, and reduced.

Understanding this definition

Using the equivalence of reduced algebras and affine sets, we have that an affine algebraic variety is an affine algebraic set such that the ideal of polynomials that defines (generates) it is radical. We can see this by taking the ring in the definition above to be the coordinate ring, with its accompanying structure sheaf.

This definition helps us see that we really are just looking at sets of solutions of polynomials, though it is very useful to look at these using functors into algebraic objects to manipulate them.

Longer formal definition (with a little less algebra)

An affine algebraic variety is a pair where

• is a topological space
• is a sub-algebra of the continuous maps to that satisfy the following property. There exists , an algebra ideal , and a homeomorphism (using the Zariski topology) such that

Note: . In other words, its is the polynomials in variables restricted to the vanishing locus.

This is a little more “coordinate free” since we just need to be able to have a space that is homeomorphic to the affine algebraic set and a subset of the functions that pull back to polynomials.

Morphism of affine algebraic varieties

Simply put, a morphism of an affine algebraic variety is a morphism of ringed spaces

the following are a few other equivalent descriptions of morphisms.

Using pullbacks

For affine varieties and , a map is a morphism if

1. is continuous

Alternates (coordinatized but more intuitive) definition

Let and be affine algebraic sets. A morphism is a function such that there exist polynomials

such that the restriction of

to is (in symbols ).

It is important to note by construction a map of a polynomials is necessarily continuous, so we just restrict them to the affine sets and .

This induces an algebra homomorphism

which is basically just composition of polynomials.

If , then

so the pre-image of closed sets is closed, so restricts to a continuous function using the subspace topology.

Notes on morphisms

An isomorphism is a morphism with an inverse morphisms. It is not enough to just check that the morphism is a bijection.

If is a morphism, then we may restrict to an algebra homomorphism

Examples

1. (they are actually isomorphic as algebras!)

3. The following morphisms

these are inverses.

4. Let be affine algebraic. Consider the (pointwise) -algebra . Then for some radical. Then we can look at

This map is surjective with (all the functions that are on ). Thus, by the first isomorphism theorem,

where denotes the coordinate ring.

Coincidence of affine variety structures

For a set with two affine algebraic variety structures and , the two structures coincide if and only if the identity map is an isomorphism.

Proof

First assume the affine algebraic variety structures are the same. This means, . Thus, if we look at the identity map

Then we can see that the pullback map

is an isomorphism because

Next, if we assume is an isomorphism then we know from class that the restriction

is an algebra isomorphism.

Thus, and are the same.

Examples

• Every vector space is an affine algebraic variety as shown here.