The basic theory of varieties in algebraic geometry

Abstract

In algebraic geometry, a variety is a set of zeroes of a set of polynomial equations in an arbitrary finite number of variables. The order reversing correspondence between varieties and ideals establishes a bridge between the algebraic nature of polynomial rings and the geometry of affine varieties. For algebraically closed fields, Hilbert's Nullstellensatz states that this order reversing correspondence restricts to a one-to-one correspondence between varieties and radical ideals, between irreducible varieties and prime ideals, and between points and maximal ideals. In this work, we shall discuss affine varieties, the Zariski topology (the topology where the closed sets are the affine varieties), coordinate rings, and morphisms between varieties.