Editing Algebraic geometry (section)

=== Affine varieties ===
{{main|Affine variety}}

First we start with a [[field (mathematics)|field]] ''k''. In classical algebraic geometry, this field was always the complex numbers '''C''', but many of the same results are true if we assume only that ''k'' is [[algebraically closed field|algebraically closed]]. We consider the [[affine space]] of dimension ''n'' over ''k'', denoted '''A'''<sup>n</sup>(''k'') (or more simply '''A'''<sup>''n''</sup>, when ''k'' is clear from the context). When one fixes a coordinate system, one may identify '''A'''<sup>n</sup>(''k'') with ''k''<sup>''n''</sup>. The purpose of not working with ''k''<sup>''n''</sup> is to emphasize that one "forgets" the vector space structure that ''k''<sup>n</sup> carries.

A function ''f'' : '''A'''<sup>''n''</sup> → '''A'''<sup>1</sup> is said to be ''polynomial'' (or ''regular'') if it can be written as a polynomial, that is, if there is a polynomial ''p'' in ''k''[''x''<sub>1</sub>,...,''x''<sub>''n''</sub>] such that ''f''(''M'') = ''p''(''t''<sub>1</sub>,...,''t''<sub>''n''</sub>) for every point ''M'' with coordinates (''t''<sub>1</sub>,...,''t''<sub>''n''</sub>) in '''A'''<sup>''n''</sup>. The property of a function to be polynomial (or regular) does not depend on the choice of a coordinate system in '''A'''<sup>''n''</sup>.

When a coordinate system is chosen, the regular functions on the affine ''n''-space may be identified with the ring of [[polynomial function]]s in ''n'' variables over ''k''. Therefore, the set of the regular functions on '''A'''<sup>''n''</sup> is a ring, which is denoted ''k''['''A'''<sup>''n''</sup>].

We say that a polynomial ''vanishes'' at a point if evaluating it at that point gives zero. Let ''S'' be a set of polynomials in ''k''['''A'''<sup>n</sup>]. The ''vanishing set of S'' (or ''vanishing locus'' or ''zero set'') is the set ''V''(''S'') of all points in '''A'''<sup>''n''</sup> where every polynomial in ''S'' vanishes. Symbolically,

:<math>V(S) = \{(t_1,\dots,t_n) \mid p(t_1,\dots,t_n) = 0 \text{ for all } p \in S\}.\,</math>

A subset of '''A'''<sup>''n''</sup> which is ''V''(''S''), for some ''S'', is called an ''algebraic set''. The ''V'' stands for ''variety'' (a specific type of algebraic set to be defined below).

Given a subset ''U'' of '''A'''<sup>''n''</sup>, can one recover the set of polynomials which generate it? If ''U'' is ''any'' subset of '''A'''<sup>''n''</sup>, define ''I''(''U'') to be the set of all polynomials whose vanishing set contains ''U''. The ''I'' stands for [[ideal (ring theory)|ideal]]: if two polynomials ''f'' and ''g'' both vanish on ''U'', then ''f''+''g'' vanishes on ''U'', and if ''h'' is any polynomial, then ''hf'' vanishes on ''U'', so ''I''(''U'') is always an ideal of the polynomial ring ''k''['''A'''<sup>''n''</sup>].

Two natural questions to ask are:
* Given a subset ''U'' of '''A'''<sup>''n''</sup>, when is ''U'' = ''V''(''I''(''U''))?
* Given a set ''S'' of polynomials, when is ''S'' = ''I''(''V''(''S''))?

The answer to the first question is provided by introducing the [[Zariski topology]], a topology on '''A'''<sup>''n''</sup> whose closed sets are the algebraic sets, and which directly reflects the algebraic structure of ''k''['''A'''<sup>''n''</sup>]. Then ''U'' = ''V''(''I''(''U'')) if and only if ''U'' is an algebraic set or equivalently a Zariski-closed set. The answer to the second question is given by [[Hilbert's Nullstellensatz]]. In one of its forms, it says that ''I''(''V''(''S'')) is the [[radical of an ideal|radical]] of the ideal generated by ''S''. In more abstract language, there is a [[Galois connection]], giving rise to two [[closure operator]]s; they can be identified, and naturally play a basic role in the theory; the [[Galois connection#Examples|example]] is elaborated at Galois connection.

For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set ''U''. [[Hilbert's basis theorem]] implies that ideals in ''k''['''A'''<sup>''n''</sup>] are always finitely generated.

An algebraic set is called ''[[irreducible component|irreducible]]'' if it cannot be written as the union of two smaller algebraic sets. Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus its elements are called the ''irreducible components'' of the algebraic set. An irreducible algebraic set is also called a ''[[algebraic variety|variety]]''. It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of a [[prime ideal]] of the [[polynomial ring]].

Some authors do not make a clear distinction between algebraic sets and varieties and use ''irreducible variety'' to make the distinction when needed.