gk-mp-9403/3

SOME CONCEPTS OF MODERN ALGEBRAIC

GEOMETRY: POINT, IDEAL AND HOMOMORPHISM

Martin Schlichenmaier

May 94

Preface

This is a write-up of lectures given at the “Kleine Herbstschule 93” of the Graduier-

tenkolleg “Mathematik im Bereich Ihrer Wechselwirkungen mit der Physik” at the

Ludwig-Maximilians-Universit¨t M¨nchen. Starting from classical algebraic geometry

a u

over the complex numbers (as it can be found for example in [GH]) it was the goal

of these lectures to introduce some concepts of the modern point of view in algebraic

geometry. Of course, it was quite impossible even to give an introduction to the whole

subject in such a limited time. For this reason the lectures and now the write-up con-

centrate on the substitution of the concept of classical points by the notion of ideals

and homomorphisms of algebras.

These concepts were established by Grothendieck in the 60s. In the following they

were proven to be very fruitful in mathematics. I do not want to give an historic

account of this claim. Let me just mention the proof of the Weil conjectures by Pierre

Deligne (see [H,App.C]) and the three more recent results: Faltings™ proof of Mordell™s

conjecture, Faltings™ proof of the Verlinde formula and Wiles™ work in direction towards

Fermat™s Last Theorem.1 But also in theoretical physics, especially in connection with

the theory of quantum groups and noncommutative geometries, it was necessary to

extend the concept of points. This is one reason for the increasing interest in modern

algebraic geometry among theoretical physicists. Unfortunately, to enter the ¬eld is

1991 Mathematics Subject Classi¬cation. 14-01, 14A15.

Key words and phrases. Algebraic Geometry.

1 At the time this is written it is not clear whether the gap found in Wiles™ “proof” really can be

closed.

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1

2 MARTIN SCHLICHENMAIER

not an easy task. It has its own very well developed language and tools. To enter it

in a linear way if it would be possible at all (which I doubt very much) would take a

prohibitive long time. The aim of the lectures was to decrease the barriers at least a

little bit and to make some appetite for further studies on a beautiful subject. I am

aiming at mathematicians and theoretical physicists who want to gain some feeling and

some understanding of these concepts. There is nothing new for algebraic geometers

here.

What are the prerequisites? I only assume some general basics of mathematics (mani-

folds, complex variables, some algebra). I try to stay elementary and hence assume only

few facts from algebraic geometry. All of these can be found in the ¬rst few chapters of

[Sch].

The write-up follows very closely the material presented at the lectures. I did with-

stand the temptation to reorganize the material to make it more systematic, to supply

all proofs, and to add other important topics. Especially the in¬nitesimal and the global

aspects are still missing. Such an extension would considerably increase the amount of

pages and hence obscure the initial goal to give a short introduction to the subject and

to make appetite for further self-study. What made it easier for me to decide in this way

is that there is a recent little book by Eisenbud and Harris available now [EH] which

(at least that is what I hope) one should be able to study with pro¬t after these lec-

tures. The book [EH] substitutes (at least partially) the for a long period only available

pedagogical introduction to the language of schemes, the famous red book of varieties

and schemes by Mumford [Mu-1].2 If you are looking for more details you can either

consult Hartshorne [H] or directly Grothendieck [EGA I],[EGA]. Of course, other good

sources are available now.

Finally, let me thank the audience for their active listening and the organizers of

the Herbstschule for the invitation. It is a pleasure for me to give special thanks to

Prof. M. Schottenloher and Prof. J. Wess.

1. Varieties

As we know from school the geometry of the plane consists of points, lines, curves,

etc. with certain relations between them. The introduction of coordinates (i.e. numbers)

to “name” the points has been proven to be very useful. In the real plane every point

can be uniquely described by its pair (±, β) of Cartesian coordinates. Here ± and β are

real numbers. Curves are “certain” subset of R — R = R2 . The notion “certain” is of

2 Which

is still very much recommended to be read. Recently, it has been reprinted in the Springer

Lecture Notes Series.

CONCEPTS OF MODERN ALGEBRAIC GEOMETRY 3

course very unsatisfactory.

In classical algebraic geometry the subsets de¬ning the geometry are the set of points

where a given set of polynomials have a common zero (if we plug in the coordinates

of the points in the polynomial). To give an example: the polynomials X and Y are

elements of the polynomial ring in 2 variables over the real numbers R. They de¬ne the

following polynomial functions:

X, Y : R2 ’ R, (±, β) ’ X(±, β) = ±, resp. Y (±, β) = β .

These two functions are called coordinate functions. The point (±0 , β0 ) ∈ R2 can be

given as zero set

{(±, β) ∈ R2 | X(±, β) ’ ±0 = 0, Y (±, β) ’ β0 = 0 } .

Let me come to the general de¬nition. For this let K be an arbitrary ¬eld (e.g.

C, R, Q, Fp , Fp , . . . ) and Kn = K — K — · · · — K the n’dimensional a¬ne space over

n times

K. I shall describe the objects of the geometry as zero sets of polynomials. For this

let Rn = K[X1 , X2 , . . . , Xn ] be the polynomial ring in n variables. A subset A of Kn

should be a geometric object if there exist ¬nitely many polynomials f1 , f2 , . . . , fs ∈ Rn

such that

x∈A f1 (x) = f2 (x) = · · · = fs (x) = 0 .

if and only if

Here and in the following it is understood that x = (x1 , x2 , . . . , xn ) ∈ Kn and f (x) ∈ K

denotes the number obtained by replacing the variable X1 by the number x1 , etc..

Using the notion of ideals it is possible to de¬ne these sets A in a more elegant

fashion. An ideal of an arbitrary ring R is a subset of R which is closed under addition :

I + I ⊆ I , and under multiplication with the whole ring: R · I ⊆ I . A good reference

to recall the necessary prerequisites from algebra is [Ku]. Now let I = (f1 , f2 , . . . , fr )

be the ideal generated by the polynomials f1 , f2 , . . . , fs which de¬ne A, e.g.

I = R · f1 + R · f2 + · · · + R · fs = {r1 f1 + r2 f2 + · · · + rs fs | ri ∈ R, i = 1, . . . , s} .

De¬nition. A subset A of Kn is called an algebraic set if there is an ideal I of Rn

such that

x ∈ A ⇐’ f (x) = 0 for all f ∈ I.

The set A is called the vanishing set of the ideal I, in symbols A = V (I) with

V (I) := { x ∈ Kn | f (x) = 0, ∀f ∈ I } . (1-1)

4 MARTIN SCHLICHENMAIER

Remark 1. It is enough to test the vanishing with respect to the generators of the ideal

in the de¬nition.

Remark 2. There is no ¬niteness condition mentioned in the de¬nition. Indeed this is

not necessary, because the polynomial ring Rn is a noetherian ring. Recall a ring is

a noetherian ring if every ideal has a ¬nite set of generators. There are other useful

equivalent de¬nitions of a noetherian ring. Let me here recall only the fact that every

strictly ascending chain of ideals (starting from one ideal) consists only of ¬nitely many

ideals. But every ¬eld K has only the (trivial) ideals {0} and K (why?), hence K

is noetherian. Trivially, all principal ideal rings (i.e. rings where every ideal can be