this assignment is compatible with the restriction of the sets where the functions are

de¬ned on. In Appendix A to this lecture you will ¬nd the exact de¬nition of a sheaf of

rings. So, given a ring R its associated a¬ne scheme is the pair (Spec(R), OR ) where

Spec(R) is the set of prime ideals made into a topological space by the Zariski topology

and OR is a sheaf of rings on Spec(R) which we will de¬ne in a minute. For simplicity

this pair is sometimes just called Spec(R).

Recall that the sets V (S) := { P ∈ Spec(R) | P ⊇ S}, where S is any S ⊆ R, are the

closed sets. Hence the sets Spec(R) \ V (S) are exactly the open sets of X := Spec(R).

There are some special open sets in X. For a single element f ∈ R we de¬ne

Xf := Spec(R) \ V (f ) = { P ∈ Spec(R) | f ∈ P } . (5-1)

The set {Xf , f ∈ R} is a basis of the topology which says that every open set is a

union of Xf . This is especially useful because the Xf are again a¬ne schemes. More

precisely, Xf = Spec(Rf ). Here the ring Rf is de¬ned as the ring of fractions with the

powers of f as denominators:

g

| g ∈ R, n ∈ N0 } .

Rf := {

fn

Let me explain this construction. It is a generalization of the way how one constructs

the rational numbers from the integers. For this let S be a multiplicative system,

i.e. a subset of R which is multiplicatively closed and contains 1. (In our example,

S := {1, f, f 2 , f 3 , . . . }.) Now introduce on the set of pairs in R — S the equivalence

relation

(t, s) ∼ (t′ , s′ ) ⇐’ ∃s′′ ∈ S such that s′′ (s′ t ’ st′ ) = 0 .

s

The equivalence class of (s, t) is denoted by . There is always a map R ’ Rf given

t

r

by r ’ . The ideals in Rf are obtained by mapping the ideals I of R to Rf and

1

multiplying them by Rf : Rf · I. By construction, f is a unit in Rf . Hence, if f ∈ P

where P is a prime ideal then Rf = Rf · P . If f ∈ P then Rf · P still is a prime ideal

of Rf . This shows Xf = Spec(Rf ). For details see [Ku].

You might ask what happens if f is nilpotent, i.e. if there is a n ∈ N such that f n = 0.

In this case f is contained in any prime ideal of Rf . Hence Spec(Rf ) = … in agreement

with Rf = {0}.

If f is not a zero divisor the map R ’ Rf is an embedding and if f is not a unit in

R the ring Rf will be bigger. This is completely in accordance with our understanding

of R resp. Rf as functions on X, resp. on the honest subset Xf . Passing from X to Xf

CONCEPTS OF MODERN ALGEBRAIC GEOMETRY 25

is something like passing from the global to the more local situation. This explains why

this process of taking the ring of fractions with respect to some multiplicative subset

S is sometimes called localization of the ring. The reader is adviced to consider the

following example. Let P be a prime ideal, show that S = R \ P is a multiplicative set.

How can one interpret the ring of fractions of R with respect to S?

Now we de¬ne our sheaf OR for the basis sets Xf . In Xf © Xg are the prime

ideals which neither contain f nor g. Hence they do not contain f · g. It follows that

Xf © Xg = Xf g . We see that the set of the Xf are closed under intersections. Note also

that X1 = X and X0 = …. We de¬ne

OR (X) := R, OR (Xf ) := Rf . (5-2)

For Xf g = Xf © Xg ⊆ Xf we de¬ne the restriction map

r

ρf g : Rf ’ (Rf )g = Rf g , r’ .

f

1

It is easy to check that all the maps ρ.. are compatible on the intersections of the basis

..

open sets. In Appendix B I will show that the other sheaf axioms are ful¬lled for the

Xf with respect to their intersections. Hence, we have de¬ned the sheaf OR on a basis

of the topology which is closed under intersections. The whole sheaf is now de¬ned by

some general construction. We set

OR (U ) := proj lim OR (Xf )

Xf ⊆U

for a general open set. For more details see [EH]. Let us collect the facts.

De¬nition. Let R be a commutative ring. The pair (Spec(R), OR ), where Spec(R)

is the space of prime ideals with the Zariski topology and OR is the sheaf of rings on

Spec(R) introduced above is called the associated a¬ne scheme Spec(R) of R. The

sheaf OR is called the structure sheaf of Spec(R).

Let me explain in which sense the elements f of an arbitrary ring R can be considered

as functions, i.e. as prescriptions how to assign a value from a ¬eld to every point. This

gives me the opportunity to introduce another important concept which is related to

points: the residue ¬elds. Fix an element f ∈ R. Let [P ] ∈ Spec(R) be a (not necessarily

closed) point, i.e. P is a prime ideal. We de¬ne

mod P ∈ R/P

f ([P ]) := f

in a ¬rst step. From the primeness of P it follows that R/P is an integral domain ring

(i.e. it contains no zero-divisor). Hence S := (R/P ) \ {0} is a multiplicative system and

26 MARTIN SCHLICHENMAIER

the ring of fractions, denoted by Quot(R/P ), is a ¬eld, the quotient ¬eld. Because R/P

is an integral domain it can be embedded into its quotient ¬eld. Hence, f ([P ]) is indeed

an element of a ¬eld. Contrary to the classical situation, if we change the point [P ] the

¬eld Quot(R/P ) will change too.

Example 1. Take again R = C[X, Y ] and f ∈ R. Here we have three di¬erent types of

points in Spec(R).

Type (i): the closed points [M ] with M = (X ’ ±, Y ’ β) a maximal ideal. We write

f = f0 + (X ’ ±) · g + (Y ’ β) · h with f0 = f (±, β) ∈ C and g, h ∈ R. Now

mod M = f0 + (X ’ ±) · g + (Y ’ β) · h

f ([M ]) = f mod M = f0 .

The quotient R/M is already a ¬eld, hence it is the residue ¬eld. In our case it is even

the base ¬eld C. The value f ([M ]) is just the value we obtain by plugging the point

(±, β) into the polynomial f . Note that the points are subvarieties of dimension 0.

Type (ii): the points [P ] with P = (h), a principal ideal. Here h is an irreducible

polynomial in the variables X and Y . If we calculate R/P we obtain C[X, Y ]/(h) which

is not a ¬eld. As residue ¬eld we obtain C(X, Y )/(h). This ¬eld consists of all rational

expressions in the variables X and Y with the relation h(X, Y ) = 0. This implies that

the transcendence degree of the residue ¬eld over the base ¬eld is one, i.e. one of the

variables X or Y is algebraically independent over C and the second variable is in an

algebraic relation with the ¬rst and the elements of C. Note that the coordinate ring

has (Krull-) dimension one and the subvariety corresponding to [P ] is a curve, i.e. is an

object of geometric dimension one.

Type (iii): [{0}] the zero ideal. In this case R/P = C[X, Y ] and the residue ¬eld is

C(X, Y ) the rational function ¬eld in two variables. In particular, its transcendence

degree is two and coincides with the (Krull-)dimensions of the coordinate ring and the

geometric dimension of the variety V ({0}) which equals the whole a¬ne plane C2 .

Strictly speaking, we have not shown (and will not do it here) that there are no other

prime ideals. But this is in fact true, see [Ku]. The equality of the transcendence degree

of the residue ¬eld and the (Krull-) dimension of the coordinate ring obtained above is

true for all varieties over arbitrary ¬elds. For example, if we replace C by R we obtain

for the closed points, the maximal ideals, either R or C as residue ¬elds. Both ¬elds

have transcendence degree 0 over R.

Example 2. Consider R = Z, the integers, then Spec(Z) consists of the zero ideal and

the principal ideals generated by prime numbers. As residue ¬eld we obtain for [0] the

¬eld Quot(Z/(0)) = Q and for the point [(p)] (which is a closed point) Fp = Z/(p),

the prime ¬eld of characteristic p. In particular, we see at this example that even for

the maximal points the residue ¬eld can vary in an essential way. Note that Z is not an

algebra over a ¬xed base ¬eld.

Up to now we considered one ring, resp. one scheme. In any category of objects one

CONCEPTS OF MODERN ALGEBRAIC GEOMETRY 27

has maps between the objects. Let ¦ : R ’ S be a ring homomorphism. If I is

any ideal of S, then ¦’1 (I) is an ideal of R. The reader is advised to check that if

P is prime then ¦’1 (P ) is again prime. Hence, ¦— : P ’ ¦’1 (P ) is a well-de¬ned

map Spec(S) ’ Spec(R). Indeed, it is even continuous because the pre-image of a

closed set is again closed. Let X = (Spec(S), OS ) and Y = (Spec(R), OR ) be two

a¬ne schemes. The map ¦ induces also a map on the level of the structure sheaves

¦— : OR ’ OS . The pair (¦— , ¦— ) of maps ful¬lls certain compatibility conditions which

makes them to a homomorphism of schemes.

We will not work with schemes in general later on but let me give at least for com-

pleteness the de¬nition here.

De¬nition. (a) A scheme is a pair X = (|X|, OX ) consisting of a topological space