non-geometry

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

(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 ) 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.

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