Xfik , k = 1, .., r is a ¬nite subcover of X. Taking for every k just one element Xjk

containing Xfik we obtain a ¬nite number of sets which is a subcover from the cover we

started with.

Note that this space is not called a compact space because the Hausdor¬ condi-

tion that every distinct two points have disjoint open neighbourhoods is obviously not

ful¬lled.

The following proposition says that the sheaf axioms (1) and (2) from App. A for the

basis open sets are ful¬lled.

Proposition. Let Xf be coverd by {Xfi }i∈I .

(a) Let g, h ∈ Rf = OR (Xf ) with g = h as elements in Rfi = OR (Xfi ) for every

i ∈ I, then g = h also in Rf .

(b) Let gi ∈ Rfi be given for all i ∈ I with gi = gj in Rfi fj , then there exist a

g ∈ Rf with g = gi in Rfi .

Proof. Because Xf = Spec(Rf ) is again an a¬ne scheme it is enough to show the

proposition for Rf = R, where R is an arbitrary ring. Let X = X fi .

i∈I

(a) Let g, h ∈ R be such that they map to the same element in Rfi . This can only

be the case if in R we have

fini · (g ’ h) = 0, ∀i ∈ I,

CONCEPTS OF MODERN ALGEBRAIC GEOMETRY 31

(see the construction of the ring of fractions above). Due to the quasicompactness it is

enough to consider ¬nitely many fi , i = 1, .., r. Hence, there is a N such that for every

i the element fiN annulates (g ’ h). There is another number M , depending on N and

r, such that we have for the following ideals

M

N N N

(f1 , f2 , . . . , fr ) ⊇ (f1 , f2 , . . . , fr ) .

Because the Xfi , i = 1, .., r are a cover of X the ideal on the right side equals (1).

Hence, also the ideal on the left. Combining 1 as linear combination of the generator

we get

N N N

1 · (g ’ h) = (c1 f1 + c2 f2 · · · + cr fr )(g ’ h) = 0 .

This shows (a)

(b) Let gi ∈ Rfi , i ∈ I be given such that gi = gj in Rfi fj . This says there as a N such

that

N N

(fi fj ) gi = (fi fj ) gj

—

gi —

in R. Note that every gi can be written as ki with gi ∈ R. Hence, if N is big enough

fi

N

the elements fi gi are in R. Again by the quasicompactness a common N will do it for

every pair (i, j). Using the same arguments as in (a) we get

ei fiN , ei ∈ R .

1=

This formula corresponds to a “partition of unity”. We set

ei fiN gi .

g=

We get

N

fj ei fiN gi =

N

ei fiN fj gj = fj gj .

N N

fj g =

i i

This shows g = gj in Rfj .

32 MARTIN SCHLICHENMAIER

6. Examples of Schemes

1. Projective Varieties. A¬ne Varieties are examples of a¬ne schemes over a ¬eld

K. They have been covered thoroughly in the other lectures. For completeness let me

mention that it is possible to introduce the projective space Pn of dimension n over

K

n+1

a ¬eld K. It can be given as orbit space (K \ {0})/ ∼, where two (n + 1)’tuple

± and β are equivalent if ± = » · β with » ∈ K, » = 0. Projective varieties are

de¬ned to be the vanishing sets of homogeneous polynomials in n + 1 variables. See for

example [Sch] for more information. What makes them so interesting is that they are

compact varieties (if K = C or R). Again everything can be dualized. One considers the

projective coordinate ring and its set of homogeneous ideals (ideals which are generated

by homogeneous elements). In the case of Pn the homogeneous coordinate ring is

K

K[Y0 , Y1 , . . . , Yn ]. Again it is possible to introduce the Zariski topology on the set of

homogeneous prime ideals. It is even possible to introduce the notion of a projective

scheme Proj, which is again a topological space together with a sheaf of rings, see [EH].

In the same way as Pn can be covered by (n + 1) a¬ne spaces Kn it is possible

K

to cover every projective scheme by ¬nitely many a¬ne schemes. This covering is even

such that the projective scheme is locally isomorphic to these a¬ne scheme. Hence, it

is a scheme. The projective scheme Proj(K[Y0 , Y1 , . . . , Yn ]) is locally isomorphic to

Spec(K[X1 , X2 , . . . , Xn ]) . For example, the open set of elements ± with Y0 (±) = 0 is

Yi

in 1-1 correspondence to it via the assignment Xi ’ .

Y0

As already said, the projective schemes are schemes and you might ask why should

one pay special attention to them. Projective schemes are quite useful. They are

schemes with rather strong additional properties. For example, in the classical case (e.g.

nonsingular varieties over C) projective varieties are compact in the classical complex

topology. This yields all the interesting results like, there are no non-constant global

analytic or harmonic functions, the theorem of Riemann-Roch is valid, the integration

is well-de¬ned, and so on. Indeed, similar results we get for projective schemes. Here it

is the feature “properness” which generalizes compactness.

2. The scheme of integers. The a¬ne scheme Spec(Z) = (Spec(Z), OSpec(Z) ) we

discussed already in the last lecture. The topological space consist of the element [{0}]

and the elements [(p)] where p takes every prime number. The residue ¬elds are Q,

resp. the ¬nite ¬elds Fp . What are the closed sets. By de¬nition, these are exactly the

sets V (S) such that there is a S ⊆ Z with

V (S) := { [(p)] ∈ Spec(Z) | (p) ⊇ S} = V ((S)) = V ((gcd(S))) .

For the last identi¬cation recall that the ideal (S) has to be generated by one element n

because Z is a principle ideal ring. Now every element in S has to be a multiple of this

CONCEPTS OF MODERN ALGEBRAIC GEOMETRY 33

n. We have to take the biggest such n which ful¬lls this condition, hence n = gcd(S).

If n = 0 then V (n) = V (0) = Spec(Z), if n = 1 then V (n) = V (1) = …, otherwise V (n)

consists of the ¬nitely many primes, resp. their ideals, dividing n. Altogether we get

that the closed sets are beside the whole space and the empty set just sets of ¬nitely

many points. As already said at some other place of these lectures Z resembles very

much K[X]. By the way, we see that the topologial closure [{0}] = Spec(Z) is the whole

space. For this reason [{0}] is called the generic point of Spec(Z).

All these has important consequences. We have two principles which can be very

useful:

(1.) Let some property be de¬ned over Z and assume it is a closed property. Assume

further that the property is true for in¬nitely many primes (e.g. the property is true if

we consider the problem in characteristic p for in¬nitely many p) then it has to be true

for the whole Spec(Z). Especially, it has to be true for all primes and for the generic

point, i.e. in characteristic zero.

(2.) Now assume that the property is an open property. If it is true for at least

one point, then it is true for all points except for possibly ¬nitely many points. In

particular, it has to be true for the generic point (characteristic zero) because every

non-empty open set has to contain the generic point.

3. A family of curve. This example illustrates the second principle above. To allow

you to make further studies by yourself on the example I take the example from [EH].

You are encouraged to develop your own examples. Consider the conic X 2 ’ Y 2 = 5.

It de¬nes a curve in the real (or complex) plane. In fact, it is already de¬ned over the

integers which says nothing more than that there is a de¬ning equation for the curve with

integer coe¬cients. Hence, it makes perfect sense to ask for points (±, β) ∈ Z2 which

solve the equation. We already saw that it is advantagous to consider the coordinate