An e-mail exchange with V.Retakh

[posic]

Lenya, privet,

I have a comment on your post on Artin-Shelter algebras. The question is
what is the right noncommutative analogue of commutative situations. I
discussed it many times with the late Gelfand. Let F be a field. Our idea
was that the correct generalization of the commutative polynomial algebra
F[x_1,...,x_n]$ is the free algebra F< x_1,...,x_n>. The growths of these
two algebras are completely different. Therefore, to look at "Algebraic
Geometry" defined by noncommutative algebras with a small growth as a
generalization of the commutative Algebraic Geometry is not so natural.
What do you think?

All the best,

Volodia

-----------------

Volodya, zdravstvujte,

the general situation is, I don't understand noncommutative geometry;
moreover, I don't think anybody does. When I was young, I was
fascinated with the subject; however, I was never able to do anything
important in it. The subject is just not developed enough; the ideology
about how it should be done is not there yet. There are all kinds of
approaches in the literature, as we know. It even looks possible
that the future will see "noncommutative geometry" as a strange
dream, typical of our age, out of which a number of unrelated subjects
will have grown, neither of them thought of as a "noncommutative
geometry" any longer.

For the time being, I would think it safe to assume that there are many
versions of noncommutative geometry, depending on the class of
noncommutative algebras being considered. There is the geometry
of noncommutative deformations of commutative objects, and there
is the geometry of noncommutative analogues of commutative objects
based on replacing the algebra of commutative polynomials with
the algebra of noncommutative ones. I have no idea how these two
are related, or which of them is deeper than the other, etc.

As a homological algebraist, I tend to see any kind of geometry as
a source of large classes of examples of homological algebra objects
whose geometric origin confirms their relevance. (Same applies to
representation theory, etc.) E.g., corings over noncommutative rings
appear in the paper of Kontsevich-Rosenberg, which for me was
an important motivation to include corings over rings on par with
semialgebras over coalgebras in what became my book on
the semi-infinite homological algebra (double-sided derived functors
of cotensor and semitensor product, etc.).

For me, the Artin-Shelter Gorenstein property is a beautiful, even if
elementary, piece of homological algebra and a topic on which
Koszul duality techniques can be tested and applied. It is potentially
open to generalizations, as the Frobenius property is itself a kind of
Gorensteinness, and so this example demonstrates that
Gorensteinness can be expected to be Koszul self-dual in some
sense. I am happy that coalgebras and the comodule-contramodule
correspondence concept provide some additional insight.

The AS Gorenstein property is also related to the Koszul property
in a nice way; and it all plays a role in the exceptional collections
and mutations business, as it was demonstrated in the paper by
Bondal and Polishchuk. It also comes up if one considers
the noncommutative Proj and tries to do Serre duality having
the analogy with the commutative projective space in mind,
as far as I remember.

The stronger condition of Artin-Shelter regularity (including
the requirement of polynomial growth) is a more complicated thing,
in the sense that it is not very clear how to use it. There is
a conjecture, formulated in our book with Sasha, that
the Hilbert series of any AS regular graded algebra is equal to
that of a certain polynomial algebra with generators in various
degrees. That is what the known d=2 and d=3 classifications
suggest. If true, this is certainly very interesting, from
the standpoint of algebras with generators and relations.

The associative algebras similar to the free noncommutative
algebra can be also distinguished homologically, but that is a very
different class. They all have homological dimension 1, to begin
with. And they are not AS Gorenstein (nor do they have polynomial
growth, of course). For a homological algebraist, the temptation
to replace "homological dimension 1" with something more
general would be almost unresistable.

Best wishes,
Lenya.

----------------

Lenya,

Thank you very much for the promt reply. I agree completely that there are
several noncommutative algebraic geometries and, may be, there *must* be
*several* such geometries. As you probably know Gelfand and I tried to do
some XIX-th centure noncommutative mathematics playing with few
fundamental problems. On the contrary, our paper with Arkady Berenstein on Lie
algebras over noncommutative rings continues the idea of noncommutative
deformations.
Unfortunetly, there is a political problem with a definition of
noncommutative geometry with its reflection on grants, influences,
invitations to conferences, etc but we cannot help it.

All the best,

Volodia

P.S. I would not mind if you decide to post our correspondence or any part
of it.

Comments

[(Anonymous)]

Provet Volodja i Lenya
Has there been any other posting of your correspondence?

спасибо
Джим
Jim Stasheff

[posic.livejournal.com]

Dear Jim,

this correspondence has not been posted anywhere else, as far as I know. My posting(s) on the Artin-Shelter algebras that the first letter refers to were written earlier on the same day and can be seen at http://posic.livejournal.com/2009/11/12/ (in Russian).

I started writing a paper about these things, but it was never finished. A very incomplete writeup is available at http://positselski.narod.ru/golodgoren.ps

Yours, Lenya