*Non-commutative geometry* seems pretty trivial compared

to commutative geometry : there are just two types of manifolds,

points and curves. However, nobody knows how to start classifying

these non-commutative curves. I do have a *conjecture* that any

non-commutative curve can (up to non-commutative birationality) be

constructed from hereditary orders over commutative curves

by *universal methods* but I’ll try to explain that another

time.

On the other hand, *non-commutative points*

have been classified (at least in principle) for at least 50

years over an arbitrary basefield $l$. *non-commutative*

$l$-*points* $P$ is an $l$-algebra such that its *double*

$d(P) = P \\otimes P^o$ ( where $P^o$ is the *opposite* algebra,

that is with the reverse multiplication) has an element$c=\\sum_i

a_i \\otimes b_i with \\sum_i a_ib_i = 1 (in $P$)$ and such that for

all p in $P$ we have that $(1 \\otimes a).c = (a \\otimes 1).c$ For

people of my generation, c is called a *separability idempotent*

and $P$ itself is called a *separable* $l$-*algebra*.

Examples of $l$-points include direct sums of full matrixrings

(of varying sizes) over $l$ or *group-algebras* $lG$ for $G$ a

finite group of n elements where n is invertible in $l$. Hence, in

particular, the group-algebra $lG$ of a p-group $G$ over a field $l$

of characteristic p is a non-commutative *singular* point and

*modular representation theory* (a theory build almost single

handed by

Richard Brauer) can be viewed as

the methods needed to resolve this singularity. Brauer’s name is

still mentioned a lot in modular representation theory, but another

of his inventions, the *Brauer group*, seems to be hardly known

among youngsters.

Still, it is a crucial tool

in classifying all non-commutative $l$-points. The algebraic

structure of an $l$-point $P$ is as follows : $$P = S_1 + S_2 + …

+ S_k$$ where each S_i is a *simple algebra* (that is, it

contains no proper twosided ideals), finite dimensional over

its center $l_i$ which is in its turn a finite dimensional

*separable* field extension of $l$. So we need to know all

simple algebras $S$, finite dimensional over their center $L$ which

is any finite dimensional separable field extension of $l$. The

algebraic structure of such an $S$ is of the form$$S = M(a,D)$$ that

is, full axa matrices with entries in $D$ where $D$ is a

*skew-field* (or some say, a *division algebra*) with

center $L$. The $L$-dimension of such a $D$ is always a square,

say b^2, so that the $L$-dimension of $S$ itself is also a square

a^2b^2. There are usually plenty such division algebras, the simplest

examples being *quaternion algebras*. Let p and q be two

non-zero elements of $L$ such that the *conic* $C : X^2-pY^2-bZ^2 =

0$ has no $L$-points in the projective $L$-plane, then the

algebra$D=(p,q)_2 = L.1 + L.i + L.j + L.ij where i^2=p, j^2=q and

ji=-ij$ is a four-dimensional skew-field over $L$. Brauer’s idea to

classify all simple $L$-algebras was to associate a group to them,

the *Brauer group*, $Br(L)$. Its elements are *equivalence*

classes of simple algebras where two simple algebras $S$ and

$S’$ are equivalent if and only if$M(m,S) = M(n,S’)$ for some sizes

m and n. Multiplication on these classes in induced by

the tensor-product (over $L$) as $S_1 \\otimes S_2$ is again a simple

$L$-algebra if $S_1$ and $S_2$ are. The Brauer group $Br(L)$ is an

Abelian *torsion* group and if we know its structure we know all

$L$-simple algebras so if we know $Br(L)$ for all finite dimensional

separable extensions $L$ of $l$ we have a full classification of

all non-commutative $l$-points.

Here are some examples

of Brauer groups : if $L$ is algebraically closed (or separable

closed), then $Br(L)=0$ so in particular, if $l$ is algebraically

closed, then the only non-commutative points are sums of matrix rings.

If $R$ is the field of real numbers, then $Br(R) = Z/2Z$ generated by

the Hamilton quaternion algebra (-1,-1)_2. If $L$ is a complete

valued number field, then $Br(L)=Q/Z$ which allows to describe also

the Brauer group of a number field in terms of its places. Brauer groups

of function fields of (commutative) varieties over an algebraically

closed basefield is usually huge but there is one noteworthy

exception $Tsen’s theorem$ which states that $Br(L)=0$ if $L$ is the

function field of a curve C over an algebraically closed field. In 1982

Merkurjev and Suslin proved a marvelous result about generators of

$Br(L)$ whenever $L$ is large enough to contain all primitive roots

of unity. They showed, in present day lingo, that $Br(L)$

is generated by non-commutative points of the *quantum-planes*

over $L$ at roots of unity. That is, it is generated by *cyclic*

algebras of the form$(p,q)_n = L

\\< X,Y>/(X^n=p,Y^n=q,YX=zXY)$where z is an n-th primitive root of

unity. Next time we will recall some basic results on the relation

between the Brauer group and *Galois cohomology*.