Editing Algebraic geometry (section)

=== Morphism of affine varieties ===
Using regular functions from an affine variety to '''A'''<sup>1</sup>, we can define [[morphism of algebraic varieties|regular map]]s from one affine variety to another. First we will define a regular map from a variety into affine space: Let ''V'' be a variety contained in '''A'''<sup>''n''</sup>. Choose ''m'' regular functions on ''V'', and call them ''f''<sub>1</sub>, ..., ''f''<sub>''m''</sub>. We define a ''regular map'' ''f'' from ''V'' to '''A'''<sup>''m''</sup> by letting {{nowrap|1=''f'' = (''f''<sub>1</sub>, ..., ''f''<sub>''m''</sub>)}}. In other words, each ''f''<sub>''i''</sub> determines one coordinate of the [[image (mathematics)|range]] of ''f''.

If ''V''′ is a variety contained in '''A'''<sup>''m''</sup>, we say that ''f'' is a ''regular map'' from ''V'' to ''V''′ if the range of ''f'' is contained in ''V''′.

The definition of the regular maps apply also to algebraic sets.
The regular maps are also called ''morphisms'', as they make the collection of all affine algebraic sets into a [[category theory|category]], where the objects are the affine algebraic sets and the [[morphism]]s are the regular maps. The affine varieties is a subcategory of the category of the algebraic sets.

Given a regular map ''g'' from ''V'' to ''V''′ and a regular function ''f'' of ''k''[''V''′], then {{nowrap|''f'' ∘ ''g'' ∈ ''k''[''V'']}}. The map {{nowrap|''f'' → ''f'' ∘ ''g''}} is a [[ring homomorphism]] from ''k''[''V''′] to ''k''[''V'']. Conversely, every ring homomorphism from ''k''[''V''′] to ''k''[''V''] defines a regular map from ''V'' to ''V''′. This defines an [[equivalence of categories]] between the category of algebraic sets and the [[dual (category theory)|opposite category]] of the finitely generated [[reduced ring|reduced]] ''k''-algebras. This equivalence is one of the starting points of [[scheme theory]].