I(A) := {f ∈ Rn | f (x) = 0, ∀x ∈ A} . (1-4)

It is the largest ideal which de¬nes A. For arbitrary ideals we always obtain I(V (I)) ⊇ I.

There is a second possibility which even takes advantage out of the non-uniqueness.

We could have added the additional data of the de¬ning ideal I in the notation. Just

simply assume that when we use A it comes with a certain I. Compare this with the

situation above where we determined the closed sets of K. Again this at the ¬rst glance

annoying fact of non-uniqueness of I will allow us to introduce multiplicities in the

following which in turn will be rather useful as we will see.

Here another warning is in order. The elements of R(A) de¬ne usual functions on

the set A. But di¬erent elements can de¬ne the same function. In particular, R(A) can

have zero divisors and nilpotent elements (which always give the zero function).

The ring R(A) contains all the geometry of A. As an example, take A to be a curve

in the plane and P a point in the plane. Then A = V ((f )) with f a polynomial in X

and Y and P = V ((X ’ ±, Y ’ β)) . Now P ‚ A (which says that the point P lies on

A) if and only if (X ’ ±, Y ’ β) ⊃ (f ). Moreover, in this case we obtain the following

8 MARTIN SCHLICHENMAIER

diagram of ring homomorphisms

/(f )

’’’

’’

R2 R(A)

¦ ¦

⊆¦ ⊆¦

/(f )

(X ’ ±, Y ’ β) ’’’

’’ (X ’ ±, Y ’ β)/(f )

¦ ¦

⊆¦ ⊆¦

’’’

’’ {0} .

(f )

The quotient (X ’ ±, Y ’ β)/(f ) is an ideal of R(A) and corresponds to the point P

lying on A.

Indeed, this is the general situation which we will study in the following sections: the

algebraic sets on A correspond to the ideals of R(A) which in turn correspond to the

ideals lying between the de¬ning ideal of A and the whole ring Rn .

Let me close this section by studying the geometry of a single point P = (±, β) ∈ K2 .

A de¬ning ideal is I = (X ’ ±, Y ’ β). If we require ”multiplicity one” this is the

de¬ning ideal. Hence, the coordinate ring R(P ) of a point is K[X, Y ]/I ∼ K . The

=

isomorphismus is induced by the homomorphism K[X, Y ] ’ K given by X ’ ±, Y ’ β.

Indeed, every element r of K[X, Y ] can be given as

r0 ∈ K, f, g ∈ K[X, Y ] .

r = r0 + (X ’ ±) · g + (Y ’ β) · f, (1-5)

Under the homomorphism r maps to r0 . Hence r is in the kernel of the map if

and only if r0 equals 0 which in turn is the case if and only if r is in the ideal I. The

description (1-5) also shows that I is a maximal ideal. We call an ideal I a maximal

ideal if there are no ideals between I and the whole ring R (and I = R). Any ideal

strictly larger than the above I would contain an r with r0 = 0. Now this ideal would

contain r, (X ’ ±), (Y ’ β) hence also r0 . Hence also (r0 )’1 · r0 = 1. But an ideal

containing 1 is always the whole ring.

On the geometric side the points are the minimal sets. On the level of the ideals

in Rn this corresponds to the fact that an ideal de¬ning a point (with multiplicity

one) is a maximal ideal. If the ¬eld K is algebraically closed then every maximal ideal

corresponds indeed to a point.

CONCEPTS OF MODERN ALGEBRAIC GEOMETRY 9

2. The spectrum of a ring

In the last lecture we saw that geometric objects are in correspondence to algebraic

objects of the coordinate ring. This we will develop more systematically in this lecture.

We had the following correspondences (1-1), (1-4)

V

’’

ideals of Rn algebraic sets

I

←’ algebraic sets.

ideals of Rn

Recall the de¬nitions: (Rn = K[X1 , X2 , . . . , Xn ])

V (I) := { x ∈ Kn | f (x) = 0, ∀f ∈ I }, I(A) := { f ∈ Rn | f (x) = 0, ∀x ∈ A } .

In general I(V (I)) will be bigger than the ideal I. Let me give an example. Consider

in C[X] the ideals I1 = (X) and I2 = (X 2 ). Then V (I1 ) = V (I2 ) = {0}. Hence both

ideals de¬ne the same point as vanishing set. Moreover I(V (I2 )) = I1 because I1 is a

maximal ideal. If we write down the coordinate ring of the two situations we obtain

for I1 the ring C[X]/(X) ∼ C. This is the expected situation because the functions on

=

a point are just the constants. For I2 we obtain C[X]/(X 2 ) ∼ C • C · « the algebra

=

2

generated by 1 and « with the relation « = 0 (X maps to «). Hence, there is no 1-1

correspondence between ideals and algebraic sets. If one wants such a correspondence

one has to throw away the ”wrong” ideals. This is in fact possible (by considering the

so called radical ideals, see the de¬nition below). Indeed, it is rather useful to allow all

ideals to obtain more general objects (which are very useful) than the classical objects.

To give an example: take the a¬ne real line and let It = (X 2 ’ t2 ) for t ∈ R be a

family of ideals. The role of t is the role of a parameter one is allowed to vary. Obviously,

It = ((X ’ t)(X + t)) = (X ’ t) · (X + t).

For t = 0 we obtain V (It ) = {t, ’t} and for t = 0 we obtain V (I0 ) = {0}. We see

that for general values of t we get two points, and for the value t = 0 one point. If we

approach with t the value 0 the two di¬erent points ±t come closer and closer together.

Now our intuition says that the limit point t = 0 better should be counted twice. This

intuition we can make mathematically precise on the level of the coordinate rings. Here

we have

Rt = R[X]/It ∼ R • R · «, «2 = t2 .

=

The coordinate ring is a two-dimensional vector space over R which re¬‚ects the fact

that we deal with two points. Everything here is also true for the exceptional value

t = 0. Especially R0 is again two-dimensional. This says we count the point {0} twice.

10 MARTIN SCHLICHENMAIER

The drawback is that the interpretation of the elements of Rt as classical functions

¯

will not be possible in all cases. In our example for t = 0 the element X will be nonzero

¯

but X(0) = 0.

For the following de¬nitions let R be an arbitrary commutative ring with unit 1.

De¬nition.

(a) An ideal P of R is called a prime ideal if P = R and a · b ∈ P implies a ∈ P or

b ∈ P.

(b) An ideal M of R is called a maximal ideal if M = R and for every ideal M ′ with

M ′ ⊇ M it follows that M ′ = M or M ′ = R.

(c) Let I be an ideal. The radical of I is de¬ned as

Rad(I) := { f ∈ R | ∃n ∈ N : f n ∈ I } .

(d) The nil radical of the ring R is de¬ned as nil(R) := Rad({0}) .

(e) A ring is called reduced if nil(R) = {0}.

(f) An ideal I is called a radical ideal if Rad(I) = I.

Starting from these de¬nitions there are a lot of easy exercises for the reader:

(1) Let P be a prime ideal. Show: R/P is a ring without zero divisor (such rings are

called integral domains).

(2) Let M be a maximal ideal. Show R/M is a ¬eld.

(3) Every maximal ideal is a prime ideal.

(4) Rad(I) is an ideal.

(5) Rad(I) equals the intersection of all prime ideals containing I.

(6) nil(R/I) = Rad(I)/I and conclude that every prime ideal is a radical ideal.

(7) Rad I is a radical ideal.

Let me return to the rings Rt de¬ned above. The ideals It are not prime because