Algebraic geometry
Morphism of affine varieties

Using regular functions from an affine variety to A, we can define regular maps 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. Choose m regular functions on V, and call them f,..., f. We define a regular map f from V to A by letting f = (f,..., f). In other words, each f determines one coordinate of the range of f.

If V ′ is a variety contained in A, 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, where the objects are the affine algebraic sets and the morphisms 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, then f ∘ g ∈ k. The map f → f ∘ g is a ring homomorphism from k to k. Conversely, every ring homomorphism from k to k defines a regular map from V to V ′. This defines an equivalence of categories between the category of algebraic sets and the opposite category of the finitely generated reduced k -algebras. This equivalence is one of the starting points of scheme theory.
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