vanishing locus

Definition

Given a set , the vanishing locus of the set , is

Note, we don’t need to do this where the base field is . We can find the vanishing locus for any family of polynomials in the polynomial ring for any algebraically closed field.

Alternate terminology: If a set is the vanishing locus of polynomials it is called an affine algebraic set.

Vanishing locus of subsets

For then . This makes sense as will have more polynomials, which imposes more conditions, and will allow for fewer 0’s

Morphisms

Let and be affine algebraic sets. A morphism is a function of of underlying sets for which there exists polynomials

which induce (via the restriction to ). That is

Note since affine algebraic varieties are defined using these algebraic sets then the morphims of affine varieties can be defined in the same way.

Vanishing Locus for ideals

Let be the ideal (ring theory) generated by . Then .

Proof

Naively, so .

Consider and

then

Therefore, and .

Importance

By this proposition, for a collection of functions, we need only think about the vanishing locus of its ideal. Or better said, for it is only necessary to look at vanishing loci of ideals.

By Hilbert Basis Theorem, the ring is Noetherian, so all ideals are finitely generated. Therefore, any vanishing locus is of the form

for finitely many polynomials (where before the set was possibly infinte.)

Going the other way

Note can be seen as a map

We can go the other direction, given an affine subset (i.e. a closed set in the affine space ) we can define the ideal

See also

• Zariski topology

• affine space

• equivalence of reduced algebras and affine sets