Tag: Quillen

In this
series of posts I’ll try to make at least part of the recent
[Kontsevich-Soibelman paper](http://www.arxiv.org/abs/math.RA/0606241) a
bit more accessible to algebraists. In non-geometry, the algebras
corresponding to *smooth affine varieties* I’ll call **qurves** (note
that they are called **quasi-free algebras** by Cuntz & Quillen and
**formally smooth** by Kontsevich). By definition, a qurve in an affine
$\mathbb{C}$-algebra A having the lifting property for algebra
maps through nilpotent ideals (extending Grothendieck’s characterization
of smooth affine algebras in the commutative case). Examples of qurves
are : finite dimensional semi-simple algebras (for example, group
algebras $\mathbb{C} G$ of finite groups), coordinate rings of
smooth affine curves or a noncommutative mixture of both, skew-group
algebras $\mathbb{C}[X] \ast G$ whenever G is a finite group of
automorphisms of the affine curve X. These are Noetherian examples but
in general a qurve is quite far from being Noetherian. More typical
examples of qurves are : free algebras $\mathbb{C} \langle x_1,\ldots,x_k \rangle$ and path algebras of finite quivers
$~\mathbb{C} Q$. Recall that a finite quiver Q s just a
directed graph and its path algebra is the vectorspace spanned by all
directed paths in Q with multiplication induced by concatenation of
paths. Out of these building blocks one readily constructs more
involved qurves via universal algebra operations such as (amalgamated)
free products, universal localizations etc. In this way, the
groupalgebra of the modular group $SL_2(\mathbb{Z})$ (as well
as that of a congruence subgroup) is a qurve and one can mix groups with
finite groupactions on curves to get qurves like $(\mathbb{C}[X] \ast G) \ast_{\mathbb{C} H} \mathbb{C} M$ whenever H is a common
subgroup of the finite groups G and M. So we have a huge class of
qurve-examples obtained from mixing finite and arithmetic groups with
curves and quivers. Qurves can we used as *machines* generating
interesting $A_{\infty}$-categories. Let us start by recalling
some facts about finite closed subschemes of an affine smooth variety Y
in the commutative case. Let **fdcom** be the category of all finite
dimensional commutative $\mathbb{C}$-algebras with morphisms
being onto algebra morphisms, then the study of finite closed subschemes
of Y is essentially the study of the covariant functor **fdcom** –>
**sets** assigning to a f.d. commutative algebra S the set of all onto
algebra maps from $\mathbb{C}[Y]$ to S. S being a f.d.
commutative semilocal algebra is the direct sum of local factors $S \simeq S_1 \oplus \ldots \oplus S_k$ where each factor has a
unique maximal ideal (a unique point in Y). Hence, our study reduces to
f.d. commutative images with support in a fixed point p of Y. But all
such quotients are also quotients of the completion of the local ring of
Y at p which (because Y is a smooth variety, say of dimension n) is
isomorphic to formal power series
$~\mathbb{C}[[x_1,\ldots,x_n]]$. So the local question, at any
point p of Y, reduces to finding all settings
$\mathbb{C}[[x_1,\ldots,x_n]] \twoheadrightarrow S \twoheadrightarrow \mathbb{C}$ Now, we are going to do something
strange (at least to an algebraist), we’re going to take duals and
translate the above sequence into a coalgebra statement. Clearly, the
dual $S^{\ast}$ of any finite dimensional commutative algebra
is a finite dimensional cocommutative coalgebra. In particular
$\mathbb{C}^{\ast} \simeq \mathbb{C}$ where the
comultiplication makes 1 into a grouplike element, that is
$\Delta(1) = 1 \otimes 1$. As long as the (co)algebra is
finite dimensional this duality works as expected : onto maps correspond
to inclusions, an ideal corresponds to a sub-coalgebra a sub-algebra
corresponds to a co-ideal, so in particular a local commutative algebra
corresponds to an pointed irreducible cocommutative coalgebra (a
coalgebra is said to be irreducible if any two non-zero subcoalgebras
have non-zero intersection, it is called simple if it has no non-zero
proper subcoalgebras and is called pointed if all its simple
subcoalgebras are one-dimensional. But what about infinite dimensional
algebras such as formal power series? Well, here the trick is not to
take all dual functions but only those linear functions whose kernel
contains a cofinite ideal (which brings us back to the good finite
dimensional setting). If one takes only those good linear functionals,
the ‘fancy’-dual $A^o$of an algebra A is indeed a coalgebra. On
the other hand, the full-dual of a coalgebra is always an algebra. So,
between commutative algebras and cocommutative coalgebras we have a
duality by associating to an algebra its fancy-dual and to a coalgebra
its full-dual (all this is explained in full detail in chapter VI of
Moss Sweedler’s book ‘Hopf algebras’). So, we can dualize the above pair
of onto maps to get coalgebra inclusions $\mathbb{C} \subset S^{\ast} \subset U(\mathfrak{a})$ where the rightmost coalgebra is
the coalgebra structure on the enveloping algebra of the Abelian Lie
algebra of dimension n (in which all Lie-elements are primitive, that is
$\Delta(x) = x \otimes 1 + 1 \otimes x$ and indeed we have that
$U(\mathfrak{a})^{\ast} \simeq \mathbb{C}[[x_1,\ldots,x_n]]$.
We have translated our local problem to finding all f.d. subcoalgebras
(containing the unique simple) of the enveloping algebra. But what is
the point of this translation? Well, we are not interested in the local
problem, but in the global problem, so we somehow have to **sum over all
points**. Now, on the algebra level that is a problem because the sum of
all local power series rings over all points is no longer an algebra,
whereas the direct sum of all pointed irreducible coalgebras $~B_Y = \oplus_{p \in Y} U(\mathfrak{a}_p)$ is again a coalgebra! That
is, we have found a huge coalgebra (which we call the coalgebra of
‘distributions’ on Y) such that for every f.d. commutative algebra S we
have $Hom_{comm alg}(\mathbb{C}[Y],S) \simeq Hom_{cocomm coalg}(S^{\ast},B_Y)$ Can we get Y back from this coalgebra of
districutions? Well, in a way, the points of Y correspond to the
group-like elements, and if g is the group-like corresponding to a point
p, we can recover the tangent-space at p back as the g-primitive
elements of the coalgebra of distributions, that is the elements such
that $\Delta(x) = x \otimes g + g \otimes x$. Observe that in
this commutative case, there are no **skew-primitives**, that is
elements such that $\Delta(x) = x \otimes g + h \otimes x$ for
different group-likes g and h. This is the coalgebra translation of the
fact that a f.d. semilocal commutative algebra is the direct sum of
local components. This is something that will definitely change if we
try to extend the above to the case of qurves (to be continued).

Now
that my non-geometry
post
string-coffee-table post
which in turn is available through a
paper

The little
this : if you have a situation where a discrete group is acting with a
bad orbit-space (for example, $GL_2(\mathbb{Z})$ acting on the whole
complex-plane, rather than just the upper half plane) you can associate
to this a $C^*$-algebra and study invariants of it and interprete them
example is the one I mentioned and where the additional noncommutative
points (coming from the orbits on the real axis) seem to contain a lot
of modular information as clarified by work of Manin&Marcolli and
Zagier. Probably the best introduction into Connes-style
non-commutative geometry
from this perspective are the Lecture on
Arithmetic Noncommutative Geometry
by Matilde Marcolli. To
algebraists : this trick is very similar to looking at the
skew-group algebra $\mathbb{C}[x_1,\ldots,x_n] * G$ if
you want to study the _orbifold_ for a finite group action on affine
space. But as algebraist we have to stick to affine varieties and
polynomials so we can only deal with the case of a finite group,
analysts can be sloppier in their functions, so they can also do
something when the group is infinite.

By the way, the
skew-group algebra idea is also why non-commutative algebraic
geometry
enters string-theory via the link with orbifolds. The
easiest (and best understood) example is that of Kleinian singularities.
The best introduction to this idea is via the Representations
of quivers, preprojective algebras and deformations of quotient
singularities
notes by Bill Crawley-Boevey.

Artin-style non-commutative geometry aka
non-commutative projective geometry originated from the
work of Artin-Tate-Van den Bergh (in the west) and Odeskii-Feigin (in
the east) to understand Sklyanin algebras associated to elliptic curves
and automorphisms via ‘geometric’ objects such as point- (and
fat-point-) modules, line-modules and the like. An excellent survey
paper on low dimensional non-commutative projective geometry is Non-commutative curves and surfaces by Toby
Stafford and
Michel Van den Bergh
. The best introduction is the (also
neverending…) book-project Non-
commutative algebraic geometry
by Paul Smith who
maintains a
noncommutative geometry and algebra resource page
page (which is

Non-geometry
started with the seminal paper ‘Algebra extensions and
nonsingularity’, J. Amer. Math. Soc. 8 (1995), 251-289 by Joachim
Cuntz
and Daniel Quillen but which is not available online. An
online introduction is Noncommutative smooth
spaces
by Kontsevich and Rosenberg. Surely, different people have
different motivations to study non-geometry. I assume Cuntz got
interested because inductive limits of separable algebras are quasi-free
(aka formally smooth aka qurves). Kontsevich and Soibelman want to study
morphisms and deformations of $A_{\infty}$-categories as they explain in
their recent
paper
. My own motivation to be interested in non-geometry is the
hope that in the next decades one will discover new exciting connections
between finite groups, algebraic curves and arithmetic groups (monstrous
moonshine
being the first, and still not entirely understood,
instance of this). Part of the problem is that these three topics seem
to be quite different, yet by taking group-algebras of finite or
arithmetic groups and coordinate rings of affine smooth curves they all
turn out to be quasi-free algebras, so perhaps non-geometry is the
unifying theory behind these seemingly unrelated topics.

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.