(Spec(R), OR ). This says that for every point x ∈ X there is an open set U containing x,

and a ring R (it may depend on the point x) such that the a¬ne scheme (Spec(R), OR )

is isomorphic to the scheme (U, OX|U ). In other words there is a homeomorphism

Ψ : U ’ Spec(R) such that there is an isomorphism of sheaves

Ψ# : OR ∼ Ψ— (OX |U ) .

=

Here the sheaf Ψ— (OX |U ) is de¬ned to be the sheaf on Spec(R) given by the assignment

Ψ— (OX |U )(W ) := OX (Ψ’1 (W )), for every open set W ⊆ Spec(R).

(b) A scheme is called an a¬ne scheme if it is globally isomorphic to an a¬ne scheme

(Spec(R), OR ) associated to a ring R.

Fact. The category of a¬ne schemes is equivalent to the category of commutative rings

with unit with the arrows (representing the maps) reversed.

There are other important concepts in this theory. First, there is the concept of a

scheme over another scheme. This is the right context to describe families of schemes.

Only within this framework it is possible to make such useful things precise as degen-

erations, moduli spaces etc. Note that every a¬ne scheme is in a natural way a scheme

over Spec(Z), because for every ring R we have the natural map Z ’ R, n ’ n · 1 .

Taking the dual map introduced above we obtain a homomorphism of schemes.

If R is a K’algebra with K a ¬eld then we have the map K ’ R, ± ’ ± · 1, which is a

ring homomorphism. Hence, we always obtain a map: Spec(R) ’ Spec(K) = ({0}, K).

By considering the coordinate ring R(V ) of an a¬ne variety V over a ¬xed algebraically

closed ¬eld K and assigning to it the a¬ne scheme Spec(R(V )) we obtain a functor

from the category of varieties over K to the category of schemes over K. The schemes

28 MARTIN SCHLICHENMAIER

corresponding to the varieties are the irreducible and reduced noetherian a¬ne schemes

of ¬nite type over Spec(K). The additional properties of the scheme are nothing else

as the corresponding properties for the de¬ning ring R(V ). Here ¬nite type means that

R(V ) is a ¬nitely generated K’algebra. You see again in which sense the schemes

extend our geometric objects from the varieties to more general “spaces”.

The second concept is the concept of a functor of points of a scheme. We saw already

at several places in the lectures that points of a geometric object can be described as

homomorphisms of the dual (algebraic) object into some simple (algebraic) object. If

X is a scheme we can associate to it the following functor from the category of schemes

to the category of sets: hX (S) = Hom(S, X). Here S is allowed to be any scheme

and Hom(S, X) is the set of homomorphisms of schemes from S to the ¬xed scheme

X. Such a homomorphism is called an S’valued point of X. Note that we are in

the geometric category, hence the order of the elements in Hom(., .) is just the other

way round compared to the former lectures. The functor hX is called the functor of

points associated to X. Now X is completely ¬xed by the functor hX . In categorical

language: X represents its own functor of points. The advantage of this view-point

is that certain questions of algebraic geometry, like the existence of a moduli space

for certain geometric data, can be easily transfered to the language of functors. One

can extract already a lot of geometric data without knowing whether there is indeed a

scheme having this functor as functor of points (i.e. representing the functor). If you

want to know more about this beautiful subject you should consult [EH] and [Mu-2].

Appendix A: The de¬nition of a sheaf of rings. A presheaf F of rings over a

topological space X assigns to every open set U in X a ring F(U ) and to every pair of

open sets V ⊆ U a homomorphism of rings

ρU : F(U ) ’ F(V ),

V

(the so called restriction map) in such a way that

ρU = id,

U

ρU —¦ ρ W = ρW for V ⊆ U ⊆ W .

V U V

Instead of ρU (f ) for f ∈ F(U ) we often use the simpler notation f|V . A presheaf is

V

called a sheaf if for every open set U and every covering (Ui ) of this open set we have

in addition:

(1) if f, g ∈ F(U ) with

f|Ui = g|Ui

for all Ui then f = g,

(2) if a set of fi ∈ F(Ui ) is given with

fi|Ui ©Uj = fj|Ui ©Uj

CONCEPTS OF MODERN ALGEBRAIC GEOMETRY 29

then there exists a f ∈ F(U ) with

f|Ui = fi .

Given two sheaves of rings F and G on X. By a sheaf homomorphism

ψ:F ’G

we understand an assignment of a ring homomorphism ψU (for every open set U )

ψU : F(U ) ’ G(U ),

which is compatible with the restriction homomorphisms

ψU

F(U ) ’’ G(U )

U

¦ ¦

¦ ¦

ψV

F(V ) ’’ G(V )

V

More information you ¬nd in [Sch].

Appendix B. The structure sheaf OR . In this appendix I like to show that the

sheaf axioms for the structure sheaf OR on X = Spec(R) are ful¬lled if we consider only

the basis open sets Xf = Spec(R) \ V (f ) . Recall that the intersection of two basis

basis open sets Xf © Xg = Xf g is again a basis open set. The sheaf OR on the basis

open sets was de¬ned to be OR (Xf ) = Rf and the restriction maps were the natural

maps

r

Rf ’ (Rf )g = Rf g , r’ .

1

Here I am following very closely the presentation in [EH].

Lemma 1. The set {Xf | f ∈ R} is a basis of the topology.

Proof. We have to show that every open set U is a union of such Xf . By de¬nition,

U = Spec(R) \ V (S) = Spec(R) \ ( (Spec(R) \ V (f )) =

V (f )) = Xf .

f ∈S f ∈S f ∈S

Obviously, only a set of generators {fi | i ∈ J} of the ideal generated by the set S

is needed. Hence, if R is a noetherian ring every open set can already be covered by

¬nitely manx Xf .

30 MARTIN SCHLICHENMAIER

Lemma 2. Let X = Spec(R) and {fi }i∈J a set of elements of R then the union of

the sets Xfi equals X if and only if the ideal generated by the fi equals the whole ring

R.

Proof. The union of the Xfi covers Spec(R) i¬ no prime ideal of R contains all the fi .

But every ideal strictly smaller than the whole ring is dominated by a maximal (and

hence prime) ideal. The above can only be the case i¬ the ideal generated by the fi is

the whole ring.

Lemma 3. The a¬ne scheme X = Spec(R) is a quasicompact space. This says every

open cover of X has a ¬nite subcover.

Proof. Let X = Xj be a cover of X. Because the basis open set Xf are a basis of

j∈J

the topology, every Xj can be given as union of Xfi . Altogether, we get a re¬nement of

the cover X = Xfi . By Lemma 2 the ideal generated by these fi is the whole ring.

i∈I

In particular, 1 is a ¬nite linear combination of the fi . Taking only these fi which occur