Here’s

an appeal to the few people working in Cuntz-Quillen-Kontsevich-whoever

noncommutative geometry (the one where smooth affine varieties

correspond to quasi-free or formally smooth algebras) : let’s rename our

topic and call it **non-geometry**. I didn’t come up with

this term, I heard in from Maxim Kontsevich in a talk he gave a couple

of years ago in Antwerp. There are some good reasons for this name

change.

The term _non-commutative geometry_ is already taken by

much more popular subjects such as _Connes-style noncommutative

differential geometry_ and _Artin-style noncommutative algebraic

geometry_. Renaming our topic we no longer have to include footnotes

(such as the one in the recent Kontsevich-Soibelman

paper) :

We use “formal” non-commutative geometry

in tensor categories, which is different from the non-commutative

geometry in the sense of Alain Connes.

or to make a

distinction between _noncommutative geometry in the small_ (which is

Artin-style) and _noncommutative geometry in the large_ (which in

non-geometry) as in the Ginzburg notes.

Besides, the stress in _non-commutative geometry_ (both in Connes-

and Artin-style) in on _commutative_. Connes-style might also be called

‘K-theory of $C^*$-algebras’ and they use the topological

information of K-theoretic terms in the commutative case as guidance to

speak about geometrical terms in the nocommutative case. Similarly,

Artin-style might be called ‘graded homological algebra’ and they

use Serre’s homological interpretation of commutative geometry to define

similar concepts for noncommutative algebras. Hence, non-commutative

geometry is that sort of non-geometry which is almost

commutative…

But the main point of naming our subject

non-geometry is to remind us not to rely too heavily on our

(commutative) geometric intuition. For example, we would expect a

manifold to have a fixed dimension. One way to define the dimension is

as the trancendence degree of the functionfield. However, from the work

of Paul Cohn (I learned about it through Aidan Schofield) we know that

quasi-free algebras usually do’nt have a specific function ring of

fractions, rather they have infinitely many good candidates for it and

these candidates may look pretty unrelated. So, at best we can define a

_local dimension_ of a noncommutative manifold at a point, say given by

a simple representation. It follows from the Cunz-Quillen tubular

neighborhood result that the local ring in such a point is of the

form

$M_n(\mathbb{C} \langle \langle z_1,\ldots,z_m \rangle

\rangle) $

(this s a noncommutative version of the classical fact

than the local ring in a point of a d-dimensional manifold is formal

power series $\mathbb{C} [[ z_1,\ldots,z_d ]] $) but in non-geometry both

m (the _local_ dimension) and n (the dimension of the simple

representation) vary from point to point. Still, one can attach to the

quasi-free algebra A a finite amount of data (in fact, a _finite_ quiver

and dimension vector) containing enough information to compute the (n,m)

couples for _all_ simple points (follows from the one quiver to rule them

all paper or see this for more

details).

In fact, one can even extend this to points

corresponding to semi-simple representations in which case one has to

replace the matrix-ring above by a ring Morita equivalent to the

completion of the path algebra of a finite quiver, the _local quiver_ at

the point (which can also be computer from the one-quiver of A. The

local coalgebras of distributions at such points of

Kontsevich&Soibelman are just the dual coalgebras of these local

algebras (in math.RA/0606241 they

merely deal with the n=1 case but no doubt the general case will appear

in the second part of their paper).

The case of the semi-simple

point illustrates another major difference between commutative geometry

and non-geometry, whereas commutative simples only have self-extensions

(so the distribution coalgebra is just the direct sum of all the local

distributions) noncommutative simples usually have plenty of

non-isomorphic simples with which they have extensions, so to get at the

global distribution coalgebra of A one cannot simply add the locals but

have to embed them in more involved coalgebras.

The way to do it

is somewhat concealed in the

third version of my neverending book (the version that most people

found incomprehensible). Here is the idea : construct a huge uncountable

quiver by taking as its vertices the isomorphism classes of all simple

A-representations and with as many arrows between the simple vertices S

and T as the dimension of the ext-group between these simples (and

again, these dimensions follow from the knowledge of the one-quiver of

A). Then, the global coalgebra of distributions of A is the limit over

all cotensor coalgebras corresponding to finite subquivers). Maybe I’ll

revamp this old material in connection with the Kontsevich&Soibelman

paper(s) for the mini-course I’m supposed to give in september.