Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations. When there is more than one variable, geometric considerations enter, and are important to understand the phenomenon. One can say that the subject starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.

The category of affine varieties

Using regular functions from an affine variety to {mathbb A}^1, we can define regular functions from one affine variety to another. First we will define a regular function from a variety into affine space: Let V be a variety contained in {mathbb A}^n. Choose m regular functions on V, and call them f1,...,fm. We define a regular function f from V to {mathbb A}^m by letting f(t1,...,tn)=(f1,...,fm). In other words, each fi determines one coordinate of the range of f.

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If V' is a variety contained in {mathbb A}^m, we say that f is a regular function from V to V' if the range of f is contained in V'.

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This makes the collection of all affine varieties into a category, where the objects are affine varieties and the morphisms are regular maps. The following theorem characterizes the category of affine varieties:

Related Topics:
Category - Morphism

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: The category of affine varieties is the opposite category to the category of finitely generated reduced k-algebras and their homomorphisms.

Related Topics:
Opposite category - Reduced - Algebras

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