Affine Varieties and Polynomial Zero Sets

If f (X) ∈k[X] define its polynomial function f ♭: kn →k by evaluation: If (a1, . . . , an) ∈kn, then f ♭: (a1, . . . , an) ↦f (a1, . . . , an). The next proposition generalizes Corollary 3.28 from one variable to several variables.

If F is a subset of k[X] = k[x1, . . . , xn], then the variety 11,12 defined by F is Var(F) = {a ∈kn : f (a) = 0 for every f (X) ∈F}; thus, Var(F) consists of all those a ∈kn which are zeros of every f (X) ∈F.

If A ⊆kn, define its coordinate ring k[A] to be the commutative ring k[A] = { f ♭|A : f (X) ∈k[X]} under pointwise operations [recall that f ♭: kn →k is the polynomial function arising from f (X)]. The polynomial f (x1, . . . , xn) = xi ∈k[X], when regarded as a polynomial function, is defined by xi : (a1, . . . , an) ↦ai; that is, xi picks out the ith coordinate of a point in kn. The reason for the name coordinate ring is that if a ∈V , then (x1(a), . . . , xn(a)) describes a. There is an obvious ring homomorphism res: k[X] →k[A], given by f (X) ↦f ♭|A, and the kernel of this restriction map is an ideal in k[X]. We will assume, from now on, that all fields k are infinite, and so we will drop the notation f ♭.

If I is an ideal in a commutative ring R, then its radical, denoted by √ I, is √ I = {r ∈R : rm ∈I for some integer m ≥1}. An ideal I is called a radical ideal 14 if √ I = I. For example, every prime ideal P is a radical ideal, for if f n ∈P, then f ∈P. Here is an example of an ideal that is not radical. Let b ∈k and let I = ((x −b)2). Now I is not a radical ideal, for (x −b)2 ∈I while x −b /∈I.

A decomposition V = V1 ∪· · · ∪Vm is an irredundant union if no Vi can be omitted; that is, for all i, V ̸= V1 ∪· · · ∪ Vi ∪· · · ∪Vm.

An ideal Q in a commutative ring R is primary if it is a proper ideal and if ab ∈Q (where a, b ∈R) and b /∈Q, then an ∈Q for some n ≥1. It is clear that every prime ideal is primary. Moreover, in Z, the ideal (pe), where p is prime and e ≥2, is a primary ideal that is not a prime ideal. Example 6.114 shows that this example is misleading: There are primary ideals that are not powers of prime ideals; there are powers of prime ideals which are not primary ideals.

A primary decomposition I = Q1 ∩. . . ∩Qr is normal if it is irredundant and if all the prime ideals Pi = √Qi are distinct.

A prime ideal P is minimal over an ideal I if I ⊆P and there is no prime ideal P′ with I ⊆P′ ⊊P.

Let k be an infinite field and let k[X] = k[x1, . . . , xn]. If f (X), g(X) ∈ k[X] satisfy f ♭= g♭, then f (x1, . . . , xn) = g(x1, . . . , xn).

Let k be a field, and let F and G be subsets of k[X]. (i) If F ⊆G ⊆k[X], then Var(G) ⊆Var(F). (ii) If F ⊆k[X] and I = (F) is the ideal generated by F, then Var(F) = Var(I).

If A ⊆kn, then there is an isomorphism k[X]/ Id(A) ∼= k[A], where k[A] is the coordinate ring of A.

If an ideal I = Id(A) for some A ⊆kn, then it is a radical ideal. Hence, the coordinate ring k[A] has no nonzero nilpotent elements.

Let k be a field and let ϕ : k[X] →k be a surjective ring homomorphism which fixes k pointwise. If J = ker ϕ, then Var(J) ̸= ∅.

Let k be an (uncountable) algebraically closed field. If I is an ideal in k[X], then Id(Var(I)) = √ I. Thus, f vanishes on Var(I) if and only if f m ∈I for some m ≥1.

(i) If V1 and V2 are varieties and Id(V1) = Id(V2), then V1 = V2. (ii) Let k be an (uncountable) algebraically closed field. If I1 and I2 are radical ideals and Var(I1) = Var(I2), then I1 = I2.

Every variety V in kn is a union of finitely many irreducible subvarieties: V = V1 ∪V2 ∪· · · ∪Vm.

Every variety V is an irredundant union of irreducible subvarieties V = V1 ∪· · · ∪Vm; moreover, the irreducible subvarieties Vi are uniquely determined by V .

If Q is a primary ideal, then its radical P = √Q is a prime ideal. Moreover, if Q is primary, then ab ∈Q and a /∈Q implies b ∈P.

Let Q be an ideal in a commutative ring R. Then Q is a primary ideal if and only if, for each a ∈R, the map aR/Q : R/Q →R/Q, given by r + Q ↦ar + Q, is either an injection or is nilpotent [(aR/Q)n = 0 for some n ≥1].

If R is a commutative ACC rings, then every proper ideal I in R has a primary decomposition.

If P is a prime ideal and Q1, . . . , Qn are P-primary ideals, then Q1 ∩· · · ∩Qn is also a P-primary ideal.

Let I be an ideal in a ACC rings R. (i) Any two normal primary decompositions of I have the same set of isolated prime ideals, and so the isolated prime ideals are uniquely determined by I. (ii) I has only finitely many minimal prime ideals. (iii) A ACC rings has only finitely many minimal prime ideals.

(i) If k is algebraically closed, then Proposition 6.90 says that if f (X) ∈k[X] is not constant, then Var( f (X)) ̸= ∅. (ii) Here are some varieties defined by two equations: Var(x, y) = {(a, b) ∈k2 : x = 0 and y = 0} = {(0, 0)} and Var(xy) = x-axis ∪y-axis. (iii) Here is an example in higher-dimensional space. Let A be an m×n matrix with entries in k. A system of m equations in n unknowns, AX = B, where B is an n × 1 column matrix, defines a variety, Var(AX = B), which is a subset of kn. Of course, AX = B is really shorthand for a set of m linear equations in n variables, and Var(AX = B) is usually called the solution set of the system AX = B; when this system is homogeneous, that is, when B = 0, then Var(AX = 0) is a subspace of kn, called the solution space of the system. ◀ The next result shows that, as far as varieties are concerned, we may just as well assume that the subsets F of k[X] are ideals of k[X]. 11There is some disagreement about the usage of this term. Some call this an affine variety, in contrast to the analogous projective variety. Some insist that varieties should be irreducible, which we will define later in this section. 12The term variety arose as a translation by E. Boi, L.

(i) Let R = Z, let (n) be a nonzero proper ideal, and let n = pe1 1 · · · pet t be the prime factorization. Then (n) = (pe1 1 ) ∩· · · ∩(pet t ) is an irredundant primary decomposition. (ii) Let R = k[x, y], where k is a field. Define Q1 = (x) and Q2 = (x, y)2. Note that Q1 is prime, and hence Q1 is P1-primary for P1 = Q1. Also, P2 = (x, y) is a maximal ideal, and so Q2 = P2 2 is P2-primary, by Proposition 6.113. Define I = Q1 ∩Q2. This primary decompostion of I is irredundant. The associated primes of I are thus {P1, P2}. ◀ There is a second uniqueness result that describes a normalized primary decomposition, but we precede it by a lemma.