Ann(M ) := { r ∈ R | r · m = 0, ∀m ∈ M } ,

the annulator of the module M . Ann(M ) is a two-sided ideal. Clearly, it is closed

under addition and is a left ideal. (This is even true for an annulator of a single element

m ∈ M ). It is also a right ideal: let s ∈ Ann(M ) and t ∈ R then (st)m = s(tm) = 0

because s annulates also tm.

CONCEPTS OF MODERN ALGEBRAIC GEOMETRY 21

De¬nition. An ideal I is called a primitive ideal if I is the annulator ideal of a simple

module M .

Let us call the set of prime, resp. primitive, resp. maximal ideals Spec(R), Priv(R)

and Max(R).

Claim.

⊇ ⊇

Spec(R) Priv(R) Max(R) .

Proof. (1). Let P be a maximal ideal. Then R/P is a (left-)module. Unfortunately, it

is not necessarily simple (as module). The submodules correspond to left-ideals lying

between P and R. Choose Q a maximal left ideal lying above P . Then R/Q is a simple

(left-) module and P · (R/Q) = 0 because P · R = P ⊆ Q. Hence, P ⊆ Ann(R/Q) and

because Ann(R/Q) is a two-sided ideal we get equality.

(2). Take P = Ann(M ), a primitive ideal. Assume P is not prime. Then there exist

a, b ∈ R but a, b ∈ P such that for all r ∈ R we get arb ∈ P . This implies arbm = 0

for all m ∈ M but bm = 0 for at least one m. Now B = R(bm) is a non-vanishing

submodule. Obviously, a ∈ Ann(B), hence B = M . This contradicts the simplicity of

M.

Clearly, in the commutative case Priv(R) = Max(R). Let me just give an example

from [GoWa] that in the noncommutative case they fall apart. Take V an in¬nite-

dimensional C’vector space. Let R be the algebra of linear endomorphisms of V and I

the nontrivial two-sided ideal consisting of linear endomorphisms with ¬nite-dimensional

image. The vector space V is an R’module by the natural action of the endomorphisms.

We get that V = R ·v where v is any non-zero vector of V . This implies that the module

V is simple and that Ann(V ) = {0}. Hence {0} is primitive, but it is not maximal

because I is lying above it.

In the commutative case we saw that we could interpret homomorphisms of the

coordinate ring (which is an algebra if we consider varieties over a base ¬eld) into a ¬eld

as points of the associated space. Indeed, it is possible to give such an interpretation

also in the noncommutative setting. Let me give an example, for details see [Ma-1].

Let Mq (2) for q ∈ C, q = 0 be the (noncommutative) C’algebra generated by a, b, c, d,

subject to the relations:

1 1 1

’ q bc,

ab = ba, ac = ca, ad = da +

q q q

(4-1)

1 1

bc = cb, bd = db, cd = dc .

q q

This algebra is constructed by ¬rst considering all possible words in a, b, c, d. This

de¬nes the free noncommutative algebra of this alphabet. Multiplication is de¬ned by

22 MARTIN SCHLICHENMAIER

concatenation of the words. Take the ideal generated by the expressions (left-side) “

(right-side) of all the relations (4-1) and build the quotient algebra. Note that for q = 1

we obtain the commutative algebra of polynomial functions on the space of all 2 — 2

matrices over C. In this sense the algebra Mq (2) represents the “quantum matrices” as

a “deformation of the usual matrices”. To end up with the quantum group Glq (2) we

would have to add another element for the formal inverse of the quantum determinant

1

D = ad ’ bc.3

q

Now let A be another algebra. We call a C’linear algebra homomorphism

Ψ ∈ Hom(Mq (2), A) an A’valued point of Mq (2). It is called a generic point if Ψ is

injective. Saying that a linear map Ψ is an algebra homomorphism is equivalent to

saying that the elements Ψ(a), Ψ(b), Ψ(c), Ψ(d) ful¬ll the same relations (4-1) as the

a, b, c and d. One might interpret Ψ as a point of the “quantum group”. But be careful,

it is only possible to “multiply” the two matrices if the images of the two maps

a1 b1 a2 b2

Ψ1 ∼ B1 := Ψ2 ∼ B2 :=

, ,

c1 d1 c2 d2

lie in a common algebra A3 , i.e. a1 , b1 , c1 , d1 ∈ A1 ⊆ A3 and a2 , b2 , c2 , d2 ∈ A2 ⊆ A3 .

Then we can multiply the two matrices B1 · B2 as prescribed by the usual matrix

product and obtain another matrix B3 with coe¬cients a3 , b3 , c3 , d3 ∈ A3 . This matrix

de¬nes only then a homomorphism of Mq (2), i.e. an A3 ’valued point if Ψ1 (Mq (2))

commutes with Ψ2 (Mq (2)) as subalgebras of A3 . In particular, the product of Ψ with

itself is not an A’valued point of Mq (2) anymore. One can show that it is an A’valued

point of Mq2 (2).

Because in the audience there a couple people who had and still have their share

in developing the fundamentals of quantum groups (the Wess-Zumino approach) there

is no need to give a lot of references on the subject. Certainly, these people know it

much better than I do. For the reader let me just quote one article by Julius Wess and

Bruno Zumino [WZ] where one ¬nds references for further study in this direction. Let

me only give the following three references of books, resp. papers of Manin which are

more connected to the theme of these lectures: “Quantum groups and noncommuta-

tive geometry” [Ma-1], “Topics in noncommutative geometry” [Ma-2], and “Notes on

quantum groups and the quantum de Rham complexes” [Ma-3].

For the general noncommutative situation I like to recommend Goodearl and War¬eld,

“An introduction to noncommutative noetherian rings” [GoWa] and Borho, Gabriel,

Rentschler, “Primideale in Einh¨llenden au¬‚¨sbarer Liealgebren” [BGR]. These books

u o

are still completely on the algebraic side of the theory. For the algebraic geometric side

there is still not very much available. Unfortunately, I am also not completely aware of

the very recent developments of the theory. The reader may use the two articles [Ar-2]

and [R] as starting points for his own exploration of the subject.

3 There are other objects which carry also the name quantum groups.

CONCEPTS OF MODERN ALGEBRAIC GEOMETRY 23

5. A¬ne schemes

Returning to the commutative setting let R be again a commutative ring with unit

1. We do not assume R to be an algebra over a ¬eld K. If we consider the theory of

di¬erentiable manifolds the model manifold is Rn . Locally any arbitrary manifold looks

like the model manifold. A¬ne schemes are the “model spaces” of algebraic geometry.

General schemes will locally look like a¬ne schemes. Contrary to the di¬erentiable

setting, there is not just one model space but a lot of them. A¬ne schemes are very

useful generalizations of a¬ne varieties. Starting from a¬ne varieties V over a ¬eld K

we saw that we were able to assign dual objects to them, the coordinate rings R(V ).

The geometric structure of V (subvarieties, points, maps, ...) are represented by the

algebraic structure of R(V ) (prime ideals, maximal ideals, ring homomorphisms, ...).

After dualization we are even able to extend our notion of “space” in the sense that we

can consider more general rings and regard them as dual objects of some generalized

“spaces”. In noncommutative quantum geometry one even studies certain noncommu-

tative algebras over a ¬eld K. Quantum spaces are the dual objects of these algebras.

We will restrict ourselves to the commutative case, but we will allow arbitrary rings.

What are the dual objects (dual to the rings) which generalize the concept of a variety.

We saw already that prime ideals of the coordinate ring correspond to subvarieties and

that closed prime ideals (at least if the ¬eld K is algebraically closed) correspond to

points. It is quite natural to take as space the set Spec(R) together with its Zariski

topology. But this is not enough. If we take for example R1 = K and R2 = K[«]/(«2 )

then in both cases Spec(Ri ) consists just of one point. It is represented in the ¬rst

case by the ideal {0} in the second case by («). Obviously, both Spec coincide. Let us

compare this with the di¬erentiable setting. For an arbitrary di¬erentiable manifold the

structure is not yet given if we consider the manifold just as a topological manifold. We

can ¬x its di¬erentiable structure if we tell what the di¬erentiable functions are. The

same is necessary in the algebraic situation. Hence, Spec(R) together with the functions

(which in the case of varieties correspond to the elements of R) should be considered as

“space”. So the space associated to a ring R should be (Spec(R), R). In fact, Spec(R)

is not a data independent of R. Nevertheless, we will write both information in view

of globalizations of the notion. Compare this again with the di¬erentiable situation. If

you have a manifold which is Rn (the model manifold) then the topology is ¬xed. But if

you have an arbitrary di¬erentiable manifold then you need a topology at the ¬rst place

to de¬ne coordinate charts at all. In view of these globalizations we additionally have to

replace the ring of functions by a data which will give us all local and global functions

together. Note that in the case of compact complex analytic manifolds there would exist

no non-constant analytic functions at all. The right setting for this is the language of

sheaves. Here it is not the time and place to introduce this language. Just let me give

you a very rough idea. A sheaf is the coding of an object which is local and global in a

24 MARTIN SCHLICHENMAIER

compatible way. A standard example (which is in some sense too simple) is the sheaf

of di¬erentiable functions on a di¬erentiable manifold X. It assigns to every open set