Affine varieties and vanishing ideals

Throughout this module k is algebraically closed unless an exception is stated; concrete worked examples use characteristic zero. Affine space $\mathbb A_k^n$ is the set of n-tuples with its polynomial functions. Given an ideal J in the polynomial ring, its zero locus consists of the tuples annihilating every element of J. It suffices to test generators because evaluation respects addition and multiplication. Hilbert's basis theorem makes a finite list sufficient even if the initial family of equations is infinite.

We call any such zero locus an affine algebraic set and reserve affine variety for a nonempty irreducible algebraic set. Some references use variety without irreducibility, so the convention matters when fraction fields appear later. Closed subsets in the Zariski topology are precisely algebraic sets. They are closed under arbitrary intersections and finite unions, but usually not arbitrary unions. A real drawing below displays only a real slice of the algebraically closed geometry, rather than replacing the ground field silently.

The set I(S) is an ideal: sums still vanish, and multiplying a vanishing polynomial by an arbitrary polynomial preserves vanishing. A larger point set imposes more tests and therefore gives a smaller vanishing ideal. Conversely, a larger equation ideal cuts out fewer possible points. These two reversals are the basic logic behind the dictionary. They prevent an easy mistake: adding equations cannot enlarge a solution set, although adding generators can make its presentation look more complicated.

Every vanishing ideal is radical. If a power of f vanishes at every point, its scalar value has zero power in a field, so that value itself is zero. Consequently the equations f and f squared have the same points even though their generated ideals need not agree. The quotient by a point set's full ideal is reduced, meaning it has no nonzero nilpotents. An arbitrary equation quotient can retain a nonzero class whose power is zero; that is additional scheme information, not an extra ordinary point.

For a closed algebraic set X, applying I and then V recovers X: its original defining equations already belong to I(X), and all of I(X) vanish on X. For an arbitrary subset S, the same operation produces its Zariski closure. An infinite subset of the affine line is dense because a nonzero univariate polynomial has only finitely many roots. Thus closure is determined by polynomial tests and can be much larger than an intuitive picture based on isolated sampled points.

Useful set identities become ideal operations. A common zero of the sum of two ideals lies in both zero loci; a zero of their product lies in their union. To prove the latter, a point outside both loci admits one nonvanishing polynomial from each ideal, whose product also does not vanish. These observations give unions of components without introducing inequalities. They also show why deleting a point generally produces an open or locally closed set rather than another closed algebraic set in the same ambient space.