# Tag: Quillen

Last time we discovered that the mental picture to view prime numbers as knots in $S^3$ was first dreamed up by David Mumford. Today, we’ll focus on where and when this happened.

3. When did Mazur write his unpublished preprint?

According to his own website, Barry Mazur did write the paper Remarks on the Alexander polynomial in 1963 or 1964. A quick look at the references gives us a coarse lower- and upper-estimate.

Apart from a paper by Iwasawa and one by Milnor, all references predate 1962 giving us a lower-bound. More interesting is reference (14) to David Mumford’s Geometric Invariant Theory (GIT) which was first published in 1965 and is referred to as ‘in preparation’, so the paper was written no later than 1965. If we look a bit closer we see than some GIT-references are very precise

indicating that Mazur must have had the final version of GIT to consult, making it rather difficult to believe that the preprint was written late 1963 or early 1964.

Mazur’s dating of the preprint is probably based on this penciled note on the frontpage of the only surviving copy of the preprint

It reads : “Date from about 63/64, H.R. Morton”. Hugh Morton of Liverpool University confirms that it is indeed his writing on the preprint.

Further, he told me that early 64 Christopher Zeeman held a Topology Symposium in Cambridge UK, where Hugh was a graduate student at the time and, as far as he could recall, Mazur attended that conference and gave him the preprint on that occasion, whence the 63/64 dating. Hugh kindly offered to double-check this with Terry Wall who cannot remember Mazur attending that particular conference.

In fact, we will see that a more correct dating of the Mazur-preprint will be : late 1964 or early 1965.

4. The birthday : July 10th 1964

Clearly, Mumford’s insight predates the Mazur-preprint. In the first section, Mazur mentions ‘Grothendieck cohomology groups’ rather than ‘Etale cohomology groups’.

At the time, Artin’s seminar notes on Grothendieck topologies (spring 1962) were widely distributed, and Artin and Grothendieck were in the process of developing etale cohomology in their Paris 1963/64 seminar SGA 4, while Mumford was working on GIT in Harvard.

Mike Artin, David Mumford and Jean-Louis Verdier all attended the Woods Hole conference from july 6 till july 31 1964, famous for producing the Atiyah-Bott fixed point theorem (according to Fulton first proved by Verdier at the conference).

Etale cohomology was a hot topic at that conference. On july 10th there were three talks, Artin spoke on ‘Etale cohomology of schemes’, Verdier on ‘A duality theorem in the etale cohomology of schemes’ and John Tate on ‘Etale cohomology over number fields’.

After a first week of talks, more informal seminars were organized, including the Atiyah-Bott seminar leading to the ‘Woods hole duality theorem’ and one by Lubin-Tate and Serre on elliptic curves and formal groups. Two seminars adressed Etale Cohomology.

Artin and Verdier ran a seminar on the etale cohomology of number fields leading to their duality result, and, three young turks : Daniel Quillen, Steve Kleiman and Robin Hartshorne ran a Baby Seminar on Etale cohomology

Probably it is safe to say that the talks by Artin, Verdier and Tate on July 10th sparked the primes=knots idea, and if not then, a couple of days later.

5. The birthplace : the Whitney Estate

The ‘Woods Hole’ conference took place at the Whitney Estate and all the lectures took place in the rustic rooms of the main building and the participants (and their families) were housed in rented cottages in the neighborhood, for the duration of the summer.

The only picture i managed to find from the Whitney house comes from a rather surprising source : Gardeners and Caretakers ofWoods Hole. Anyway, here it is :

Probably, the knots=primes analogy was first dreamed up inside, or in the immediate neighborhood, on a walk to or from the cottages, overlooking the harbor.

The Monster is the largest of the 26 sporadic simple groups and has order

808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000

= 2^46 3^20 5^9 7^6 11^2 13^3 17 19 23 29 31 41 47 59 71.

It is not so much the size of its order that makes it hard to do actual calculations in the monster, but rather the dimensions of its smallest non-trivial irreducible representations (196 883 for the smallest, 21 296 876 for the next one, and so on).

In characteristic two there is an irreducible representation of one dimension less (196 882) which appears to be of great use to obtain information. For example, Robert Wilson used it to prove that The Monster is a Hurwitz group. This means that the Monster is generated by two elements g and h satisfying the relations

$g^2 = h^3 = (gh)^7 = 1$

Geometrically, this implies that the Monster is the automorphism group of a Riemann surface of genus g satisfying the Hurwitz bound 84(g-1)=#Monster. That is,

g=9619255057077534236743570297163223297687552000000001=42151199 * 293998543 * 776222682603828537142813968452830193

Or, in analogy with the Klein quartic which can be constructed from 24 heptagons in the tiling of the hyperbolic plane, there is a finite region of the hyperbolic plane, tiled with heptagons, from which we can construct this monster curve by gluing the boundary is a specific way so that we get a Riemann surface with exactly 9619255057077534236743570297163223297687552000000001 holes. This finite part of the hyperbolic tiling (consisting of #Monster/7 heptagons) we’ll call the empire of the monster and we’d love to describe it in more detail.

Look at the half-edges of all the heptagons in the empire (the picture above learns that every edge in cut in two by a blue geodesic). They are exactly #Monster such half-edges and they form a dessin d’enfant for the monster-curve.

If we label these half-edges by the elements of the Monster, then multiplication by g in the monster interchanges the two half-edges making up a heptagonal edge in the empire and multiplication by h in the monster takes a half-edge to the one encountered first by going counter-clockwise in the vertex of the heptagonal tiling. Because g and h generated the Monster, the dessin of the empire is just a concrete realization of the monster.

Because g is of order two and h is of order three, the two permutations they determine on the dessin, gives a group epimorphism $C_2 \ast C_3 = PSL_2(\mathbb{Z}) \rightarrow \mathbb{M}$ from the modular group $PSL_2(\mathbb{Z})$ onto the Monster-group.

In noncommutative geometry, the group-algebra of the modular group $\mathbb{C} PSL_2$ can be interpreted as the coordinate ring of a noncommutative manifold (because it is formally smooth in the sense of Kontsevich-Rosenberg or Cuntz-Quillen) and the group-algebra of the Monster $\mathbb{C} \mathbb{M}$ itself corresponds in this picture to a finite collection of ‘points’ on the manifold. Using this geometric viewpoint we can now ask the question What does the Monster see of the modular group?

To make sense of this question, let us first consider the commutative equivalent : what does a point P see of a commutative variety X?

Evaluation of polynomial functions in P gives us an algebra epimorphism $\mathbb{C}[X] \rightarrow \mathbb{C}$ from the coordinate ring of the variety $\mathbb{C}[X]$ onto $\mathbb{C}$ and the kernel of this map is the maximal ideal $\mathfrak{m}_P$ of
$\mathbb{C}[X]$ consisting of all functions vanishing in P.

Equivalently, we can view the point $P= \mathbf{spec}~\mathbb{C}[X]/\mathfrak{m}_P$ as the scheme corresponding to the quotient $\mathbb{C}[X]/\mathfrak{m}_P$. Call this the 0-th formal neighborhood of the point P.

This sounds pretty useless, but let us now consider higher-order formal neighborhoods. Call the affine scheme $\mathbf{spec}~\mathbb{C}[X]/\mathfrak{m}_P^{n+1}$ the n-th forml neighborhood of P, then the first neighborhood, that is with coordinate ring $\mathbb{C}[X]/\mathfrak{m}_P^2$ gives us tangent-information. Alternatively, it gives the best linear approximation of functions near P.
The second neighborhood $\mathbb{C}[X]/\mathfrak{m}_P^3$ gives us the best quadratic approximation of function near P, etc. etc.

These successive quotients by powers of the maximal ideal $\mathfrak{m}_P$ form a system of algebra epimorphisms

$\ldots \frac{\mathbb{C}[X]}{\mathfrak{m}_P^{n+1}} \rightarrow \frac{\mathbb{C}[X]}{\mathfrak{m}_P^{n}} \rightarrow \ldots \ldots \rightarrow \frac{\mathbb{C}[X]}{\mathfrak{m}_P^{2}} \rightarrow \frac{\mathbb{C}[X]}{\mathfrak{m}_P} = \mathbb{C}$

and its inverse limit $\underset{\leftarrow}{lim}~\frac{\mathbb{C}[X]}{\mathfrak{m}_P^{n}} = \hat{\mathcal{O}}_{X,P}$ is the completion of the local ring in P and contains all the infinitesimal information (to any order) of the variety X in a neighborhood of P. That is, this completion $\hat{\mathcal{O}}_{X,P}$ contains all information that P can see of the variety X.

In case P is a smooth point of X, then X is a manifold in a neighborhood of P and then this completion
$\hat{\mathcal{O}}_{X,P}$ is isomorphic to the algebra of formal power series $\mathbb{C}[[ x_1,x_2,\ldots,x_d ]]$ where the $x_i$ form a local system of coordinates for the manifold X near P.

Right, after this lengthy recollection, back to our question what does the monster see of the modular group? Well, we have an algebra epimorphism

$\pi~:~\mathbb{C} PSL_2(\mathbb{Z}) \rightarrow \mathbb{C} \mathbb{M}$

and in analogy with the commutative case, all information the Monster can gain from the modular group is contained in the $\mathfrak{m}$-adic completion

$\widehat{\mathbb{C} PSL_2(\mathbb{Z})}_{\mathfrak{m}} = \underset{\leftarrow}{lim}~\frac{\mathbb{C} PSL_2(\mathbb{Z})}{\mathfrak{m}^n}$

where $\mathfrak{m}$ is the kernel of the epimorphism $\pi$ sending the two free generators of the modular group $PSL_2(\mathbb{Z}) = C_2 \ast C_3$ to the permutations g and h determined by the dessin of the pentagonal tiling of the Monster’s empire.

As it is a hopeless task to determine the Monster-empire explicitly, it seems even more hopeless to determine the kernel $\mathfrak{m}$ let alone the completed algebra… But, (surprise) we can compute $\widehat{\mathbb{C} PSL_2(\mathbb{Z})}_{\mathfrak{m}}$ as explicitly as in the commutative case we have $\hat{\mathcal{O}}_{X,P} \simeq \mathbb{C}[[ x_1,x_2,\ldots,x_d ]]$ for a point P on a manifold X.

Here the details : the quotient $\mathfrak{m}/\mathfrak{m}^2$ has a natural structure of $\mathbb{C} \mathbb{M}$-bimodule. The group-algebra of the monster is a semi-simple algebra, that is, a direct sum of full matrix-algebras of sizes corresponding to the dimensions of the irreducible monster-representations. That is,

$\mathbb{C} \mathbb{M} \simeq \mathbb{C} \oplus M_{196883}(\mathbb{C}) \oplus M_{21296876}(\mathbb{C}) \oplus \ldots \ldots \oplus M_{258823477531055064045234375}(\mathbb{C})$

with exactly 194 components (the number of irreducible Monster-representations). For any $\mathbb{C} \mathbb{M}$-bimodule $M$ one can form the tensor-algebra

$T_{\mathbb{C} \mathbb{M}}(M) = \mathbb{C} \mathbb{M} \oplus M \oplus (M \otimes_{\mathbb{C} \mathbb{M}} M) \oplus (M \otimes_{\mathbb{C} \mathbb{M}} M \otimes_{\mathbb{C} \mathbb{M}} M) \oplus \ldots \ldots$

and applying the formal neighborhood theorem for formally smooth algebras (such as $\mathbb{C} PSL_2(\mathbb{Z})$) due to Joachim Cuntz (left) and Daniel Quillen (right) we have an isomorphism of algebras

$\widehat{\mathbb{C} PSL_2(\mathbb{Z})}_{\mathfrak{m}} \simeq \widehat{T_{\mathbb{C} \mathbb{M}}(\mathfrak{m}/\mathfrak{m}^2)}$

where the right-hand side is the completion of the tensor-algebra (at the unique graded maximal ideal) of the $\mathbb{C} \mathbb{M}$-bimodule $\mathfrak{m}/\mathfrak{m}^2$, so we’d better describe this bimodule explicitly.

Okay, so what’s a bimodule over a semisimple algebra of the form $S=M_{n_1}(\mathbb{C}) \oplus \ldots \oplus M_{n_k}(\mathbb{C})$? Well, a simple S-bimodule must be either (1) a factor $M_{n_i}(\mathbb{C})$ with all other factors acting trivially or (2) the full space of rectangular matrices $M_{n_i \times n_j}(\mathbb{C})$ with the factor $M_{n_i}(\mathbb{C})$ acting on the left, $M_{n_j}(\mathbb{C})$ acting on the right and all other factors acting trivially.

That is, any S-bimodule can be represented by a quiver (that is a directed graph) on k vertices (the number of matrix components) with a loop in vertex i corresponding to each simple factor of type (1) and a directed arrow from i to j corresponding to every simple factor of type (2).

That is, for the Monster, the bimodule $\mathfrak{m}/\mathfrak{m}^2$ is represented by a quiver on 194 vertices and now we only have to determine how many loops and arrows there are at or between vertices.

Using Morita equivalences and standard representation theory of quivers it isn’t exactly rocket science to determine that the number of arrows between the vertices corresponding to the irreducible Monster-representations $S_i$ and $S_j$ is equal to

$dim_{\mathbb{C}}~Ext^1_{\mathbb{C} PSL_2(\mathbb{Z})}(S_i,S_j)-\delta_{ij}$

Now, I’ve been wasting a lot of time already here explaining what representations of the modular group have to do with quivers (see for example here or some other posts in the same series) and for quiver-representations we all know how to compute Ext-dimensions in terms of the Euler-form applied to the dimension vectors.

Right, so for every Monster-irreducible $S_i$ we have to determine the corresponding dimension-vector $~(a_1,a_2;b_1,b_2,b_3)$ for the quiver

$\xymatrix{ & & & & \vtx{b_1} \\ \vtx{a_1} \ar[rrrru]^(.3){B_{11}} \ar[rrrrd]^(.3){B_{21}} \ar[rrrrddd]_(.2){B_{31}} & & & & \\ & & & & \vtx{b_2} \\ \vtx{a_2} \ar[rrrruuu]_(.7){B_{12}} \ar[rrrru]_(.7){B_{22}} \ar[rrrrd]_(.7){B_{23}} & & & & \\ & & & & \vtx{b_3}}$

Now the dimensions $a_i$ are the dimensions of the +/-1 eigenspaces for the order 2 element g in the representation and the $b_i$ are the dimensions of the eigenspaces for the order 3 element h. So, we have to determine to which conjugacy classes g and h belong, and from Wilson’s paper mentioned above these are classes 2B and 3B in standard Atlas notation.

So, for each of the 194 irreducible Monster-representations we look up the character values at 2B and 3B (see below for the first batch of those) and these together with the dimensions determine the dimension vector $~(a_1,a_2;b_1,b_2,b_3)$.

For example take the 196883-dimensional irreducible. Its 2B-character is 275 and the 3B-character is 53. So we are looking for a dimension vector such that $a_1+a_2=196883, a_1-275=a_2$ and $b_1+b_2+b_3=196883, b_1-53=b_2=b_3$ giving us for that representation the dimension vector of the quiver above $~(98579,98304,65663,65610,65610)$.

Okay, so for each of the 194 irreducibles $S_i$ we have determined a dimension vector $~(a_1(i),a_2(i);b_1(i),b_2(i),b_3(i))$, then standard quiver-representation theory asserts that the number of loops in the vertex corresponding to $S_i$ is equal to

$dim(S_i)^2 + 1 – a_1(i)^2-a_2(i)^2-b_1(i)^2-b_2(i)^2-b_3(i)^2$

and that the number of arrows from vertex $S_i$ to vertex $S_j$ is equal to

$dim(S_i)dim(S_j) – a_1(i)a_1(j)-a_2(i)a_2(j)-b_1(i)b_1(j)-b_2(i)b_2(j)-b_3(i)b_3(j)$

This data then determines completely the $\mathbb{C} \mathbb{M}$-bimodule $\mathfrak{m}/\mathfrak{m}^2$ and hence the structure of the completion $\widehat{\mathbb{C} PSL_2}_{\mathfrak{m}}$ containing all information the Monster can gain from the modular group.

But then, one doesn’t have to go for the full regular representation of the Monster. Any faithful permutation representation will do, so we might as well go for the one of minimal dimension.

That one is known to correspond to the largest maximal subgroup of the Monster which is known to be a two-fold extension $2.\mathbb{B}$ of the Baby-Monster. The corresponding permutation representation is of dimension 97239461142009186000 and decomposes into Monster-irreducibles

$S_1 \oplus S_2 \oplus S_4 \oplus S_5 \oplus S_9 \oplus S_{14} \oplus S_{21} \oplus S_{34} \oplus S_{35}$

(in standard Atlas-ordering) and hence repeating the arguments above we get a quiver on just 9 vertices! The actual numbers of loops and arrows (I forgot to mention this, but the quivers obtained are actually symmetric) obtained were found after laborious computations mentioned in this post and the details I’ll make avalable here.

Anyone who can spot a relation between the numbers obtained and any other part of mathematics will obtain quantities of genuine (ie. non-Inbev) Belgian beer…

If you ever sat through a lecture by Alain Connes you will know about his insistence on the ‘canonical dynamic nature of noncommutative manifolds’. If you haven’t, he did write a blog post Heart bit 1 about it.

I’ll try to explain here that there is a definite “supplément d’âme” obtained in the transition from classical (commutative) spaces to the noncommutative ones. The main new feature is that “noncommutative spaces generate their own time” and moreover can undergo thermodynamical operations such as cooling, distillation etc…

Here a section from his paper A view of mathematics :

Indeed even at the coarsest level of understanding of a space provided by measure
theory, which in essence only cares about the “quantity of points” in a space, one
ﬁnds unexpected completely new features in the noncommutative case. While it
had been long known by operator algebraists that the theory of von-Neumann
algebras represents a far reaching extension of measure theory, the main surprise
which occurred at the beginning of the seventies is that such an algebra M
inherits from its noncommutativity a god-given time evolution:

$\delta~:~\mathbb{R} \rightarrow Out(M)$

where $Out M = Aut M/Int M$ is the quotient of the group of automorphisms of M
by the normal subgroup of inner automorphisms. This led in my thesis to the
reduction from type III to type II and their automorphisms and eventually to the
classiﬁcation of injective factors.

Even a commutative manifold has a kind of dynamics associated to it. Take a suitable vectorfield, consider the flow determined by it and there’s your ‘dynamics’, or a one-parameter group of automorphisms on the functions. Further, other classes of noncommutative algebras have similar features. For example, Cuntz and Quillen showed that also formally smooth algebras (the noncommutative manifolds in the algebraic world) have natural Yang-Mills flows associated to them, giving a one-parameter subgroup of automorphisms.

Let us try to keep far from mysticism and let us agree that by ‘time’ (let alone ‘god given time’) we mean a one-parameter subgroup of algebra automorphisms of the noncommutative algebra. In nice cases, such as some von-Neumann algebras this canonical subgroup is canonical in the sense that it is unique upto inner automorphisms.

In the special case of the Bost-Connes algebra these automorphisms $\sigma_t$ are given by $\sigma_t(X_n) = n^{it} X_n$ and $\sigma_t(Y_{\lambda}) = Y_{\lambda}$.

This one-parameter subgroup is crucial in the definition of the so called KMS-states (for Kubo-Martin and Schwinger) which is our next goal.

Yesterday, Ed Segal gave a talk at the Arts. His title “Superpotential algebras from 3-fold singularities” didnt look too promising to me. And sure enough it was all there again : stringtheory, D-branes, Calabi-Yaus, superpotentials, all the pseudo-physics babble that spreads virally among the youngest generation of algebraists and geometers.

Fortunately, his talk did contain a general ringtheoretic gem. After a bit of polishing up this gem, contained in his paper The A-infinity Deformation Theory of a Point and the Derived Categories of Local Calabi-Yaus, can be stated as follows.

Let $A$ be a $\mathbb{C}$-algebra and let $M = S_1 \oplus \ldots \oplus S_k$ be a finite dimensional semi-simple representation with distinct simple components. Let $\mathfrak{m}$ be the kernel of the algebra epimorphism $A \rightarrow S$ to the semi-simple algebra $S=End(M)$. Then, the $\mathfrak{m}$-adic completion of $A$ is Morita-equivalent to the completion of a quiver-algebra with relations. The nice thing is that both the quiver and relations come in a canonical way from the $A_{\infty}$-structure on the Ext-algebra $Ext^{\bullet}_A(M,M)$. More precisely, there is an isomorphism

$\hat{A}_{\mathfrak{m}} \simeq \frac{\hat{T}_S(Ext^1_A(M,M)^{\ast})}{(Im(HMC)^{\ast})}$

where the homotopy Maurer-Cartan map comes from the $A_{\infty}$ structure maps

$HMC = \oplus_i m_i~:~T_S(Ext_A^1(M,M)) \rightarrow Ext^2_A(M,M)$

and hence the defining relations of the completion are given by the image of the dual of this map.

For ages, Ive known this result in the trivial case of formally smooth algebras (where $Ext^2_A(M,M)=0$ and hence there are no relations to divide out) and where it is a consequence of a special case of the Cuntz-Quillen “tubular neighborhood” result. Completions of formally smooth algebras at semi-simples are Morita equivalent to completions of path algebras. This fact motivated all the local-quiver technology that was developed here in Antwerp over the last decade (see my book if you want to know the details).

Also for 3-dimensional Calabi-Yau algebras it states that the completions at semi-simples are Morita equivalent to completions of quotients of path algebras by the relations coming from a superpotential (aka a necklace) by taking partial noncommutative derivatives. Here the essential ingredient is that $Ext^2_A(M,M)^{\ast} \simeq Ext^1_A(M,M)$ in this case.

The
problem with criticizing others is that you have to apply the same
standards to your own work. So, as of this afternoon, I do agree with
all those who said so before : my book is completely unreadable and
should either be dumped or entirely rewritten!

Here’s what happened :
Last week I did receive the contract to publish _noncommutative
geometry@n_ in a reputable series. One tiny point though, the editors
felt that the title was somewhat awkward and would stand out with
respect to the other books in the series, so they proposed as an
alternative title _Noncommutative Geometry_. A tall order, I thought,
but then, if others are publishing books with such a title why
shouldn’t I do the same?

The later chapters are quite general, anyway,
and if I would just spice them up a little adding recent material it
might even improve the book. So, rewriting two chapters and perhaps
adding another “motivational chapter” aimed at physicists… should
be doable in a month, or two at the latest which would fit in nicely
with the date the final manuscript is due.

This week, I got myself once
again in writing mode : painfully drafting new sections at a pace of 5
to 6 pages a day. Everything was going well. Today I wanted to finish
the section on the “one quiver to rule them all”-trick and was
already mentally planning the next section in which I would give details
for groups like $PSL_2(\mathbb{Z})$ and $GL_2(\mathbb{Z})$, all I
needed was to type in a version of the proof of the last proposition.

The proof uses a standard argument, which clearly should be in the book
so I had to give the correct reference and started browsing through the
could not find it!???_ And, it was not just some minor technical lemma,
but a result which is crucial to the book’s message (for the few who
want to know, the result is the construction and properties of the local
quiver at a semi-simple representation of a Quillen-smooth algebra). Of
course, there is a much more general result contained in the book, but
you have to be me (or have to be drilled by me) to see the connection…
Not good at all! I’d better sleep on this before taking further
steps

Last time we
have seen that the _coalgebra of distributions_ of an affine smooth
variety is the direct sum (over all points) of the dual to the etale
local algebras which are all of the form $\mathbb{C}[[ x_1,\ldots,x_d ]]$ where $d$ is the dimension of the
variety. Generalizing this to _non-commutative_ manifolds, the first
questions are : “What is the analogon of the power-series algebra?” and
do all ‘points’ of our non-commutative manifold do have such local
algebras? Surely, we no longer expect the variables to commute, so a
non-commutative version of the power series algebra should be
$\mathbb{C} \langle \langle x_1,\ldots,x_d \rangle \rangle$,
the ring of formal power series in non-commuting variables. However,
there is still another way to add non-commutativity and that is to go
from an algebra to matrices over the algebra. So, in all we would expect
to be our _local algebras_ at points of our non-commutative manifold to
be isomorphic to $M_n(\mathbb{C} \langle \langle x_1,\ldots,x_d \rangle \rangle)$ As to the second question : _qurves_ (that is,
the coordinate rings of non-commutative manifolds) do have such algebras
as local rings provided we take as the ‘points’ of the non-commutative
variety the set of all _simple_ finite dimensional representations of
the qurve. This is a consequence of the _tubular neighborhood theorem_
due to [Cuntz](http://wwwmath.uni-muenster.de/u/cuntz/cuntz.html) and
[Quillen](http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Quillen.html). In more details : If A is a qurve, then a simple
$n$-dimensional representation corresponds to an epimorphism
$\pi~:~A \rightarrow S = M_n(\mathbb{C})$ and if we take
$\mathfrak{m}=Ker(\pi)$, then
$M=\mathfrak{m}/\mathfrak{m}^2$ is an $S$-bimodule and
the $\mathfrak{m}$-adic completion of A is isomorphic to the
completed tensor-algebra $\hat{T}_S(M) \simeq M_n(\mathbb{C} \langle \langle x_1,\ldots,x_d \rangle \rangle)$ In contrast with
the commutative case however where the dimension remains constant over
all points, here the numbers n and d can change from simple to simple.
For n this is clear as it gives the dimension of the simple
representation, but also d changes (it is the local dimension of the
variety classifying simple representations of the same dimension). Here
an easy example : Consider the skew group algebra $A = \mathbb{C}[x] \star C_2$ with the action given by sending $x \mapsto -x$. Then A is a qurve and its center is
$\mathbb{C}[y]$ with $y=x^2$. Over any point $y \not= 0$ there is a unique simple 2-dimensional representation of A
giving the local algebra $M_2(\mathbb{C}[[y]])$. If
$y=0$ the situation is more complicated as the local structure
of A is given by the algebra $\begin{bmatrix} \mathbb{C}[[y]] & \mathbb{C}[[y]] \\ (y) & \mathbb{C}[[y]] \end{bmatrix}$ So, over
this point there are precisely 2 one-dimensional simple representations
corresponding to the maximal ideals $\mathfrak{m}_1 = \begin{bmatrix} (y) & \mathbb{C}[[y]] \\ (y) & \mathbb{C}[[y]] \end{bmatrix}~\qquad \text{and}~\qquad \mathfrak{m}_2 = \begin{bmatrix} \mathbb{C}[[y]] & \mathbb{C}[[y]] \\ (y) & (y) \end{bmatrix}$ and
both ideals are idempotent, that is $\mathfrak{m}_i^2 = \mathfrak{m}_i$ whence the corresponding bimodule $M_i = 0$ so the local algebra in either of these two points is just
$\mathbb{C}$. Ok, so the comleted local algebra at each point
is of the form $M_n(\mathbb{C}\langle \langle x_1,\ldots,x_d \rangle \rangle)$, but what is the corresponding dual coalgebra. Well,
$\mathbb{C} \langle \langle x_1,\ldots,x_d \rangle \rangle$ is
the algebra dual to the _cofree coalgebra_ on $V = \mathbb{C} x_1 + \ldots + \mathbb{C}x_d$. As a vectorspace this is the
tensor-algebra $T(V) = \mathbb{C} \langle x_1,\ldots,x_d \rangle$ with the coalgebra structure induced by the bialgebra
structure defined by taking all varaibales to be primitives, that is
$\Delta(x_i) = x_i \otimes 1 + 1 \otimes x_i$. That is, the
coproduct on a monomial gives all different expressions $m_1 \otimes m_2$ such that $m_1m_2 = m$. For example,
$\Delta(x_1x_2) = x_1x_2 \otimes 1 + x_1 \otimes x_2 + 1 \otimes x_1x_2$. On the other hand, the dual coalgebra of
$M_n(\mathbb{C})$ is the _matrix coalgebra_ which is the
$n^2$-dimensional vectorspace $\mathbb{C}e_{11} + \ldots + \mathbb{C}e_{nn}$ with comultiplication $\Delta(e_{ij}) = \sum_k e_{ik} \otimes e_{kj}$ The coalgebra corresponding to the
local algebra $M_n(\mathbb{C}\langle \langle x_1,\ldots,x_d \rangle \rangle)$ is then the tensor-coalgebra of the matrix coalgebra and
the cofree coalgebra. Having obtained the coalgebra at each point
(=simple representation) of our noncommutative manifold one might think
that the _coalgebra of non-commutative distributions_ should be the
direct sum of all this coalgebras, summed over all points, as in the
commutative case. But then we would forget about a major difference
between the commutative and the non-commutative world : distinct simples
can have non-trivial extensions! The mental picture one might have
about simples having non-trivial extensions is that these points lie
‘infinitesimally close’ together. In the $\mathbb{C}[x] \star C_2$ example above, the two one-dimensional simples have
non-trivial extensions so they should be thought of as a cluster of two
infinitesimally close points corresponding to the point $y=0$
(that is, this commutative points splits into two non-commutative
points). Btw. this is the reason why non-commutative algebras can be
used to resolve commutative singularities (excessive tangents can be
split over several non-commutative points). While this is still pretty
harmless when the algebra is finite over its center (as in the above
example where only the two one-dimensionals have extensions), the
situation becomes weird over general qurves as ‘usually’ distinct
simples have non-trivial extensions. For example, for the free algebra
$\mathbb{C}\langle x,y \rangle$ this is true for all simples…
So, if we want to continue using this image of points lying closely
together this immediately means that non-commutative ‘affine’ manifolds
behave like compact ones (in fact, it turns out to be pretty difficult
to ‘glue’ together qurves into ‘bigger’ non-commutative manifolds, apart
from the quiver examples of [this old
paper](http://www.arxiv.org/abs/math.AG/9907136)). So, how to bring
this new information into our coalgebra of distributions? Well, let’s
repeat the previous argument not with just one point but with a set of
finitely many points. Then we have a _semi-simple algebra_ quotient
$\pi~:~A \rightarrow S = M_{n_1}(\mathbb{C}) \oplus \ldots \oplus M_{n_k}(\mathb{C})$ and taking again
$\mathfrak{m}=Ker(\pi)$ and
$M=\mathfrak{m}/\mathfrak{m}^2$, then $M$ is again an
S-bimodule. Now, any S-bimodule can be encoded into a _quiver_ Q on k
points, the number of arrows from vertex i to vertex j being the number
of components in M of the form $M_{n_i \times n_j}(\mathbb{C})$. Again, it follows from the tubular neighborhood
theorem that the $\mathfrak{m}$-adic completion of A is
isomorphic to the completion of an algebra Morita equivalent to the
_path algebra_ $\mathbb{C} Q$ (being the tensor algebra
$T_S(M)$). As all the local algebras of the points are
quotients of this quiver-like completion, on the coalgebra level our
local coalgebras will be sub coalgebras of the coalgebra which is
co-Morita equivalent (and believe it or not but coalgebraists have a
name for this : _Takeuchi equivalence_) to the _quiver coalgebra_ which
is the vectorspace of the path algebra $\mathbb{C} Q$ with
multiplication induced by making all arrows from i to j skew-primitives,
that is, $\Delta(a) = e_i \otimes a + a \otimes e_j$ where the
$e_i$ are group-likes corresponding to the vertices. If all of
ths is a bit too much co to take in at once, I suggest the paper by Bill
Chin [A brief introduction to coalgebra representation
theory](http://condor.depaul.edu/~wchin/crt.pdf#search=%22%22A%20brief%20introduction%20to%20coalgebra%20representation%20theory%22%22). The
_coalgebra of noncommutative distributions_ we are after at is now the
union of all these Takeuchi-equivalent quiver coalgebras. In easy
examples such as the $\mathbb{C}[x] \star C_2$-example this
coalgebra is still pretty small (the sum of the local coalgebras
corresponding to the local algebras $M_2(\mathbb{C}[[x]])$
summed over all points $y \not= 0$ summed with the quiver
coalgebra of the quiver $\xymatrix{\vtx{} \ar@/^/[rr] & & \vtx{} \ar@/^/[ll]}$ In general though this is a huge object and we would
like to have a recipe to construct it from a manageable _blue-print_ and
that is what we will do next time.

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.

A couple of days ago Ars Mathematica had a post Cuntz on noncommutative topology pointing to a (new, for me) paper by Joachim Cuntz

A couple of years ago, the Notices of the AMS featured an article on noncommutative geometry a la Connes: Quantum Spaces and Their Noncommutative Topology by Joachim Cuntz. The hallmark of this approach is the heavy reliance on K theory. The first few pages of the article are fairly elementary (and full of intriguing pictures), before the K theory takes over.

A few comments are in order. To begin, the paper is **not** really about noncommutative geometry a la Connes, but rather about noncommutative geometry a la Cuntz&Quillen (based on quasi-free algebras) or, equivalently, a la Kontsevich (formally smooth algebras) or if I may be so bold a la moi (qurves).

About the **intruiging pictures** : it seems to be a recent trend in noncommutative geometry research papers to include meaningless pictures to lure the attention of the reader. But, unlike aberrations such as the recent pastiche by Alain Connes and Mathilde Marcolli A Walk in the Noncommutative Garden, Cuntz is honest about their true meaning

I am indebted to my sons, Nicolas and Michael,
for the illustrations to the examples above. Since
these pictures have no technical meaning, they
are only meant to provide a kind of suggestive
visualization of the corresponding quantum spaces.

As one of these pictures made it to the cover of the **Notices** an explanation was included by the cover-editor

The image on this month’s cover arose from
Joachim Cuntz’s effort to render into visible art
his own internal vision of a noncommutative
torus, an object otherwise quite abstract. His
original idea was then implemented by his son
Michael in a program written in Pascal. More
explicitly, he says that the construction started
out with a triangle in a square, then translated
the triangle by integers times a unit along a line
with irrational slope; plotted the images thus
obtained in a periodic manner; and stopped
just before the figure started to seem cluttered.
Many mathematicians carry around inside
their heads mental images of the abstractions
they work with, and manipulate these objects
somehow in conformity with their mental imagery. They probably also make aesthetic judgements of the value of their work according to
the visual qualities of the images. These presumably common phenomena remain a rarely
explored domain in either art or psychology.

—Bill Casselman(covers@ams.org)

There can be no technical meaning to the pictures as in the Connes and Cuntz&Quillen approach there is only a noncommutative algebra and _not_ an underlying geometric space, so there is no topology, let alone a noncommutative topology. Of course, I do understand why Cuntz&others name it as such. They view the noncommutative algebra as the ring of functions on some virtual noncommutative space and they compute topological invariants (such as K-groups) of the algebras and interprete them as information about the noncommutative topology of these virtual and unspecified spaces.

Still, it is perfectly possible to associate to a qurve (aka quasi-free algebra or formally smooth algebra) a genuine noncommutative topological space. In this series of posts I’ll explain the little I know of the history of this topic, the thing I posted about it a couple of years ago, why I abandoned the project and the changes I made to it since and the applications I have in mind, both to new problems (such as the birational_classification of qurves) as well as classical problems (such as rationality problems for $PGL_n$ quotient spaces).

Although others have tried to define noncommutative topologies before, I learned about them from Fred Van Oystaeyen. Fred spend the better part of his career constructing structure sheaves associated to noncommutative algebras, mainly to prime Noetherian algebras (the algebras of preference for the majority of non-commutative algebraists). So, suppose you have an ordinary (meaning, the usual commutative definition) topological space X associated to this algebra R, he wants to define an algebra of sections on every open subset $X(\sigma)$ by taking a suitable localization of the algebra $Q_{\sigma}(R)$. This localization is taken with respect to a suitable filter of left ideals $\mathcal{L}(\sigma)$ of R and is defined to be the subalgebra of the classiocal quotient ring $Q(R)$ (which exists because $R$ is prime Noetherian in which case it is a simple Artinian algebra)

$Q_{\sigma}(R) = { q \in Q(R)~|~\exists L \in \mathcal{L}(\sigma)~:~L q \subset R }$

(so these localizations are generalizations of the usual Ore-type rings of fractions). But now we come to an essential point : if we want to glue this rings of sections together on an intersection $X(\sigma) \cap X(\tau)$ we want to do this by ‘localizing further’. However, there are two ways to do this, either considering $~Q_{\sigma}(Q_{\tau}(R))$ or considering $Q_{\tau}(Q_{\sigma}(R))$ and these two algebras are only the same if we impose fairly heavy restrictions on the filters (or on the algebra) such as being compatible.

As this gluing property is essential to get a sheaf of noncommutative algebras we seem to get stuck in the general (non compatible) case. Fred’s way out was to make a distinction between the intersection $X_{\sigma} \cap X_{\tau}$ (on which he put the former ring as its ring of sections) and the intersection $X_{\tau} \cap X_{\sigma}$ (on which he puts the latter one). So, the crucial new ingredient in a noncommutative topology is that the order of intersections of opens matter !!!

Of course, this is just the germ of an idea. He then went on to properly define what a noncommutative topology (and even more generally a noncommutative Grothendieck topology) should be by using this localization-example as guidance. I will not state the precise definition here (as I will have to change it slightly later on) but early version of it can be found in the Antwerp Ph.D. thesis by Luc Willaert (1995) and in Fred’s book Algebraic geometry for associative algebras.

Although _qurves_ are decidedly non-Noetherian (apart from trivial cases), one can use Fred’s idea to associate a noncommutative topological space to a qurve as I will explain next time. The quick and impatient may already sneak at my old note a non-commutative topology on rep A but please bear in mind that I changed my mind since on several issues…