equivalence of reduced algebras and affine sets

An important idea in algebraic geometry is to take geometric objects such points or sets in an affine space and think of them as algebraic objects such as rings or ideals.

Bijection for affine spaces

There exists a bijection

where denotes the vanishing locus of and is the ideal (ring theory) of polynomials that vanish on .

Equivalence of categories

The bijection above can be stated even stronger. For the categories (with an algebraically closed field)

• • Objects: reduced finitely generated -algebras
  • Morphisms: -algebra morphisms
• • Objects: affine algebraic sets in (for all ).
  • Morphisms: Morphisms of affine algebraic sets

There is an equivalence of categories

for the opposite category of .

In picture form:

Note that reduced finitely generated algebras are better to work with since they are self-contained and are intrinsic. Affine algebraic sets are not intrinsic, they can always be embedded in a bigger space.