affine algebraic variety

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.

Short formal definition

An affine algebraic variety is an affine algebraic set such that the ideal of polynomials that defines (generates) it is prime.

This definition helps me see that we really are just looking at sets of solutions.

Longer formal definition

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 β€œcoordinate 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

For affine varieties and , a map is a morphism if

  1. is continuous

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, the we may restrict to an algebra homomorphism

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.