2. Krull dimension of a ring 3

Theorem 1.8 (Hilbert's Nullstellensatz). Let R — K[#i,... ,xn] be a poly-

nomial ring over an algebraically closed field K. If Var(a) = Var(b) for

ideals a, b C R, then rad a = rad b.

Consequently the map a \— Var(a) is a containment-reversing bisection

between radical ideals of K.[x\^... ,xn] and algebraic sets in

Kn.

D

For a proof, solve [6, Problem 7.14]. The following corollary, also referred

to as the Nullstellensatz, tells us when polynomial equations have a solution.

Corollary 1.9. Let R — K[xi,... ,xn] be a polynomial ring over an alge-

braically closed field K. Then polynomials / i , . . . , fm G R have a common

zero if and only if (/i,... , fm) ^ R.

Proof. Since Var(i?) = 0, an ideal a of R satisfies Var(a) = 0 if and only

if rad a = i?, which happens if and only if a = R. •

Corollary 1.10. Let R — K[xi,... , xn] be a polynomial ring over an alge-

braically closed field K. Then the maximal ideals of R are the ideals

(x\ — a\,..., xn — an), where cti G IK.

Proof. Let m be a maximal ideal of R. Then m ^ i?, so Corollary 1.9

implies that Var(m) contains a point (ai,... , an) of

Kn.

But then

Var(xi — a i , . . . , xn — an) C Var(m),

so radm C rad(xi — a i , . . . ,xn — an). Since m and {pc\ — a i , . . . ,x

n

— an)

are maximal ideals, it follows that they must be equal. •

Exercise 1.11. This is also taken from [95]. Prove that if K is not alge-

braically closed, any algebraic set in

Kn

is the zero set of a single polynomial.

2. Krull dimension of a ring

We would like a notion of dimension for algebraic sets which agrees with

the vector space dimension if the algebraic set is a vector space and gives

a suitable generalization of the inequality in Example 1.2. The situation

is certainly more complicated than with vector spaces. For example, not

all points of an algebraic set have similar neighborhoods—the algebraic set

defined by xy — 0 and xz = 0 is the union of a line and a plane.

A good theory of dimension requires some notions from commutative

algebra, which we now proceed to recall.

Definition 1.12. Let R be a ring. The spectrum of i?, denoted Speci?, is

the set of prime ideals of R with the Zariski topology^ which is the topology

where the closed sets are

V(a) = {p G Speci? | a C p} for ideals a C R.