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Tag: Brauer

THE rationality problem

Published April 27, 2007 by lievenlb

This morning, Esther Beneish
arxived the paper The center of the generic algebra of degree p that may contain the most
significant advance in my favourite problem for over 15 years! In it she
claims to prove that the center of the generic division algebra of
degree p is stably rational for all prime values p. Let me begin by
briefly explaining what the problem is all about. Consider one n by n
matrix A which is sufficiently general, then it will have all its
eigenvalues distinct, but then it is via the Jordan normal form theorem uniquely
determined upto conjugation (that is, base change) by its
characteristic polynomial. In
other words, the conjugacy class of a sufficiently general n by n matrix
depends freely on the coefficients of the characteristic polynomial
(which are the n elementary symmetric functions in the eigenvalues of
the matrix). Now what about couples of n by n matrices (A,B) under
simultaneous conjugation (that is all couples of the form $~(g A
g^{-1}, g B g^{-1}) $ for some invertible n by n matrix g) ??? So,
does there exist a sort of Jordan normal form for couples of n by n
matrices which are sufficiently general? That is, are there a set of
invariants for such couples which determine it is freely upto
simultaneous conjugation?

For couples of 2 by 2 matrices, Claudio Procesi rediscovered an old
result due to James Sylvester saying
that this is indeed the case and that the set of invariants consists of
the five invariants Tr(A),Tr(B),Det(A),Det(B) and Tr(AB). Now, Claudio
did a lot more in his paper. He showed that if you could prove this for
couples of matrices, you can also do it for triples, quadruples even any
k-tuples of n by n matrices under simultaneous conjugation. He also
related this problem to the center of the generic division algebra of
degree n (which was introduced earlier by Shimshon Amitsur in a rather
cryptic manner and for a while he simply refused to believe Claudio’s
description of this division algebra as the one generated by two
_generic_ n by n matrices, that is matrices filled with independent
variables). Claudio also gave the description of the center of this
algebra as a field of lattice-invariants (over the symmetric group S(n)
) which was crucial in subsequent investigations. If you are interested
in the history of this problem, its connections with Brauer group
problems and invariant theory and a short description of the tricks used
in proving the results I’ll mention below, you might have a look at the
talk Centers of Generic Division Algebras, the rationality problem 1965-1990
I gave in Chicago in 1990.

The case of couples of 3 by 3 matrices was finally
settled in 1979 by Ed Formanek and a
year later he was able to solve also the case of couples of 4 by 4
matrices in a fabulous paper. In it, he used solvability of S(4) in an
essential way thereby hinting at the possibility that the problem might
no longer have an affirmative answer for larger values of n. When I read
his 4×4 paper I believed that someone able to prove such a result must
have an awesome insight in the inner workings of matrices and decided to
dedicate myself to this problem the moment I would get a permanent
job… . But even then it is a reckless thing to do. Spending all of
your time to such a difficult problem can be frustrating as there is no
guarantee you’ll ever write a paper. Sure, you can find translations of
the problem and as all good problems it will have connections with other
subjects such as moduli spaces of vectorbundles and of quiver
representations, but to do the ‘next number’ is another matter.

Fortunately, early 1990, together with
Christine Bessenrodt we were
able to do the next two ‘prime cases’ : couples of 5 by 5 and couples of
7 by 7 matrices (Katsylo and Aidan Schofield had already proved that if
you could do it for couples of k by k and l by l matrices and if k and l
were coprime then you could also do it for couples of kl by kl matrices,
so the n=6 case was already done). Or did we? Well not quite, our
methods only allowed us to prove that the center is stably rational
that is, it becomes rational by freely adjoining extra variables. There
are examples known of stably rational fields which are NOT rational, but
I guess most experts believe that in the case of matrix-invariants
stable rationality will imply rationality. After this paper both
Christine and myself decided to do other things as we believed we had
reached the limits of what the lattice-method could do and we thought a
new idea was required to go further. If today’s paper by Esther turns
out to be correct, we were wrong. The next couple of days/weeks I’ll
have a go at her paper but as my lattice-tricks are pretty rusty this
may take longer than expected. Still, I see that in a couple of weeks
there will be a meeting in
Atlanta were Esther
and all experts in the field will be present (among them David Saltman
and Jean-Louis Colliot-Thelene) so we will know one way or the other
pretty soon. I sincerely hope Esther’s proof will stand the test as she
was the only one courageous enough to devote herself entirely to the
problem, regardless of slow progress.

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anabelian geometry

Published March 22, 2007 by lievenlb

Last time we saw
that a curve defined over $\overline{\mathbb{Q}} $ gives rise
to a permutation representation of $PSL_2(\mathbb{Z}) $ or one
of its subgroups $\Gamma_0(2) $ (of index 2) or
$\Gamma(2) $ (of index 6). As the corresponding
monodromy group is finite, this representation factors through a normal
subgroup of finite index, so it makes sense to look at the profinite
completion of $SL_2(\mathbb{Z}) $, which is the inverse limit
of finite
groups $\underset{\leftarrow}{lim}~SL_2(\mathbb{Z})/N $
where N ranges over all normalsubgroups of finite index. These
profinte completions are horrible beasts even for easy groups such as
$\mathbb{Z} $. Its profinite completion
is

$\underset{\leftarrow}{lim}~\mathbb{Z}/n\mathbb{Z} =
\prod_p \hat{\mathbb{Z}}_p $

where the right hand side
product of p-adic integers ranges over all prime numbers! The
_absolute Galois group_
$G=Gal(\overline{\mathbb{Q}}/\mathbb{Q}) $ acts on all curves
defined over $\overline{\mathbb{Q}} $ and hence (via the Belyi
maps ans the corresponding monodromy permutation representation) there
is an action of $G $ on the profinite completions of the
carthographic groups.

This is what Grothendieck calls anabelian
algebraic geometry

Returning to the general
case, since finite maps can be interpreted as coverings over
$\overline{\mathbb{Q}} $ of an algebraic curve defined over
the prime field $~\mathbb{Q} $ itself, it follows that the
Galois group $G $ of $\overline{\mathbb{Q}} $ over
$~\mathbb{Q} $ acts on the category of these maps in a
natural way.
For instance, the operation of an automorphism
$~\gamma \in G $ on a spherical map given by the rational
function above is obtained by applying $~\gamma $ to the
coefficients of the polynomials P , Q. Here, then, is that
mysterious group $G $ intervening as a transforming agent on
topologico- combinatorial forms of the most elementary possible
nature, leading us to ask questions like: are such and such oriented
maps ‚conjugate or: exactly which are the conjugates of a given
oriented map? (Visibly, there is only a finite number of these).
I considered some concrete cases (for coverings of low degree) by
various methods, J. Malgoire considered some others ‚ I doubt that
there is a uniform method for solving the problem by computer. My
reflection quickly took a more conceptual path, attempting to
apprehend the nature of this action of G.
One sees immediately
that roughly speaking, this action is expressed by a certain
outer action of G on the profinite com- pactification of the
oriented cartographic group $C_+^2 = \Gamma_0(2) $ , and this
action in its turn is deduced by passage to the quotient of the
canonical outer action of G on the profinite fundamental group
$\hat{\pi}_{0,3} $ of
$(U_{0,3})_{\overline{\mathbb{Q}}} $ where
$U_{0,3} $ denotes the typical curve of genus 0 over the
prime field Q, with three points re- moved.
This is how my
attention was drawn to what I have since termed anabelian
algebraic geometry, whose starting point was exactly a study
(limited for the moment to characteristic zero) of the action of
absolute Galois groups (particularly the groups Gal(K/K),
where K is an extension of finite type of the prime field) on
(profinite) geometric fundamental groups of algebraic varieties
(defined over K), and more particularly (break- ing with a
well-established tradition) fundamental groups which are very far
from abelian groups (and which for this reason I call
anabelian).
Among these groups, and very close to
the group $\hat{\pi}_{0,3} $ , there is the profinite
compactification of the modular group $Sl_2(\mathbb{Z}) $,
whose quotient by its centre ±1 contains the former as congruence
subgroup mod 2, and can also be interpreted as an oriented
cartographic group, namely the one classifying triangulated
oriented maps (i.e. those whose faces are all triangles or
monogons).

and a bit further, on page
250

I would like to conclude this rapid outline
with a few words of commentary on the truly unimaginable richness
of a typical anabelian group such as $SL_2(\mathbb{Z}) $
doubtless the most remarkable discrete infinite group ever
encountered, which appears in a multiplicity of avatars (of which
certain have been briefly touched on in the present report), and which
from the point of view of Galois-Teichmuller theory can be
considered as the fundamental ‚building block‚ of the
Teichmuller tower
The element of the structure of
$Sl_2(\mathbb{Z}) $ which fascinates me above all is of course
the outer action of G on its profinite compactification. By
Bielyi’s theorem, taking the profinite compactifications of subgroups
of finite index of $Sl_2(\mathbb{Z}) $, and the induced
outer action (up to also passing to an open subgroup of G), we
essentially find the fundamental groups of all algebraic curves (not
necessarily compact) defined over number fields K, and the outer
action of $Gal(\overline{K}/K) $ on them at least it is
true that every such fundamental group appears as a quotient of one
of the first groups.
Taking the anabelian yoga
(which remains conjectural) into account, which says that an anabelian
algebraic curve over a number field K (finite extension of Q) is
known up to isomorphism when we know its mixed fundamental group (or
what comes to the same thing, the outer action of
$Gal(\overline{K}/K) $ on its profinite geometric
fundamental group), we can thus say that
all algebraic
curves defined over number fields are contained in the profinite
compactification $\widehat{SL_2(\mathbb{Z})} $ and in the
knowledge of a certain subgroup G of its group of outer
automorphisms!

To study the absolute
Galois group $Gal(\overline{\mathbb{\mathbb{Q}}}/\mathbb{Q}) $ one
investigates its action on dessins denfants. Each dessin will be part of
a finite family of dessins which form one orbit under the Galois action
and one needs to find invarians to see whether two dessins might belong
to the same orbit. Such invariants are called _Galois invariants_ and
quite a few of them are known.

Among these the easiest to compute
are

  • the valency list of a dessin : that is the valencies of all
    vertices of the same type in a dessin
  • the monodromy group of a dessin : the subgroup of the symmetric group $S_d $ where d is
    the number of edges in the dessin generated by the partitions $\tau_0 $
    and $\tau_1 $ For example, we have seen
    before     that the two
    Mathieu-dessins

form a Galois orbit. As graphs (remeber we have to devide each
of the edges into two and the midpoints of these halfedges form one type
of vertex, the other type are the black vertices in the graphs) these
are isomorphic, but NOT as dessins as we have to take the embedding of
them on the curve into account. However, for both dessins the valency
lists are (white) : (2,2,2,2,2,2) and (black) :
(3,3,3,1,1,1) and one verifies that both monodromy groups are
isomorphic to the Mathieu simple group $M_{12} $ though they are
not conjugated as subgroups of $S_{12} $.

Recently, new
Galois invariants were obtained from physics. In Children’s drawings
from Seiberg-Witten curves
the authors argue that there is a close connection between Grothendiecks
programme of classifying dessins into Galois orbits and the physics
problem of classifying phases of N=1 gauge theories…

Apart
from curves defined over $\overline{\mathbb{Q}} $ there are
other sources of semi-simple $SL_2(\mathbb{Z}) $
representations. We will just mention two of them and may return to them
in more detail later in the course.

Sporadic simple groups and
their representations There are 26 exceptional finite simple groups
and as all of them are generated by two elements, there are epimorphisms
$\Gamma(2) \rightarrow S $ and hence all their representations
are also semi-simple $\Gamma(2) $-representations. In fact,
looking at the list of ‘standard generators’ of the sporadic
simples

(here the conjugacy classes of the generators follow the
notation of the Atlas project) we see that all but
possibly one are epimorphic images of $\Gamma_0(2) = C_2 \ast
C_{\infty} $ and that at least 12 of then are epimorphic images
of $PSL_2(\mathbb{Z}) = C_2 \ast
C_3 $.

Rational conformal field theories Another
source of $SL_2(\mathbb{Z}) $ representations is given by the
modular data associated to rational conformal field theories.

These
representations also factor through a quotient by a finite index normal
subgroup and are therefore again semi-simple
$SL_2(\mathbb{Z}) $-representations. For a readable
introduction to all of this see chapter 6 \”Modular group
representations throughout the realm\” of the
book Moonshine beyond the monster the bridge connecting algebra, modular forms and physics by Terry
Gannon. In fact, the whole book
is a good read. It introduces a completely new type of scientific text,
that of a neverending survey paper…

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permutation representations of monodromy groups

Published March 20, 2007 by lievenlb

Today we will explain how curves defined over
$\overline{\mathbb{Q}} $ determine permutation representations
of the carthographic groups. We have seen that any smooth projective
curve $C $ (a Riemann surface) defined over the algebraic
closure $\overline{\mathbb{Q}} $ of the rationals, defines a
_Belyi map_ $\xymatrix{C \ar[rr]^{\pi} & & \mathbb{P}^1} $ which is only ramified over the three points
$\\{ 0,1,\infty \\} $. By this we mean that there are
exactly $d $ points of $C $ lying over any other point
of $\mathbb{P}^1 $ (we call $d $ the degree of
$\pi $) and that the number of points over $~0,1~ $ and
$~\infty $ is smaller than $~d $. To such a map we
associate a _dessin d\’enfant_, a drawing on $C $ linking the
pre-images of $~0 $ and $~1 $ with exactly $d $
edges (the preimages of the open unit-interval). Next, we look at
the preimages of $~0 $ and associate a permutation
$\tau_0 $ of $~d $ letters to it by cycling
counter-clockwise around these preimages and recording the edges we
meet. We repeat this procedure for the preimages of $~1 $ and
get another permutation $~\tau_1 $. That is, we obtain a
subgroup of the symmetric group $ \langle \tau_0,\tau_1
\rangle \subset S_d $ which is called the monodromy
group of the covering $\pi $.

For example, the
dessin on the right is
associated to a degree $8 $ map $\mathbb{P}^1 \rightarrow
\mathbb{P}^1 $ and if we let the black (resp. starred) vertices be
the preimages of $~0 $ (respectively of $~1 $), then the
corresponding partitions are $\tau_0 = (2,3)(1,4,5,6) $
and $\tau_1 = (1,2,3)(5,7,8) $ and the monodromy group
is the alternating group $A_8 $ (use
GAP ).

But wait! The map is also
ramified in $\infty $ so why don\’t we record also a
permutation $\tau_{\infty} $ and are able to compute it from
the dessin? (Note that all three partitions are needed if we want to
reconstruct $C $ from the $~d $ sheets as they encode in
which order the sheets fit together around the preimages). Well,
the monodromy group of a $\mathbb{P}^1 $ covering ramified only
in three points is an epimorphic image of the fundamental
group of the sphere
minus three points $\pi_1(\mathbb{P}^1 – { 0,1,\infty
}) $ That is, the group of all loops beginning and
ending in a basepoint upto homotopy (that is, two such loops are the
same if they can be transformed into each other in a continuous way
while avoiding the three points).

This group is generated by loops
$\sigma_i $ running from the basepoint to nearby the i-th
point, doing a counter-clockwise walk around it and going back to be
basepoint $Q_0 $ and the epimorphism to the monodromy group is given by sending

$\sigma_1 \mapsto \tau_0~\quad~\sigma_2 \mapsto
\tau_1~\quad~\sigma_3 \mapsto \tau_{\infty} $

Now,
these three generators are not independent. In fact, this fundamental
group is

$\pi_1(\mathbb{P}^1 – \\{ 0,1,\infty \\}) =
\langle \sigma_1,\sigma_2,\sigma_3~\mid~\sigma_1 \sigma_2
\sigma_3 = 1 \rangle $

To understand this, let us begin
with an easier case, that of the sphere minus one point. The fundamental group of the plane minus one point is
$~\mathbb{Z} $ as it encodes how many times we walk around the
point. However, on the sphere the situation is different as we can make
our walk around the point longer and longer until the whole walk is done
at the backside of the sphere and then we can just contract our walk to
the basepoint. So, there is just one type of walk on a sphere minus one
point (upto homotopy) whence this fundamental group is trivial. Next,
let us consider the sphere minus two points


Repeat the foregoing to the walk $\sigma_2 $, that
is, strech the upper part of the circular tour all over the backside of
the sphere and then we see that we can move it to fit with the walk
$\sigma_1$ BUT for the orientation of the walk! That is, if we do this
modified walk $\sigma_1 \sigma_2^{\’} $ we just made the
trivial walk. So, this fundamental group is $\langle
\sigma_1,\sigma_2~\mid~\sigma_1 \sigma_2 = 1 \rangle =
\mathbb{Z} $ This is also the proof of the above claim. For,
we can modify the third walk $\sigma_3 $ continuously so that
it becomes the walk $\sigma_1 \sigma_2 $ but
with the reversed orientation ! As $\sigma_3 =
(\sigma_1 \sigma_2)^{-1} $ this allows us to compute the
\’missing\’ permutation $\tau_{\infty} = (\tau_0
\tau_1)^{-1} $ In the example above, we obtain
$\tau_{\infty}= (1,2,6,5,8,7,4)(3) $ so it has two cycles
corresponding to the fact that the dessin has two regions (remember we
should draw ths on the sphere) : the head and the outer-region. Hence,
the pre-images of $\infty$ correspond to the different regions of the
dessin on the curve $C $. For another example,
consider the degree 168 map

$K \rightarrow \mathbb{P}^1 $

which is the modified orbit map for the action of
$PSL_2(\mathbb{F}_7) $ on the Klein quartic.
The corresponding dessin is the heptagonal construction of the Klein
quartic

Here, the pre-images of 1 correspond to the midpoints of the
84 edges of the polytope whereas the pre-images of 0 correspond to the
56 vertices. We can label the 168 half-edges by numbers such that
$\tau_0 $ and $\tau_1 $ are the standard generators b
resp. a of the 168-dimensional regular representation (see the atlas
page ).
Calculating with GAP the element $\tau_{\infty} = (\tau_0
\tau_1)^{-1} = (ba)^{-1} $ one finds that this permutation
consists of 24 cycles of length 7, so again, the pre-images of
$\infty $ lie one in each of the 24 heptagonal regions of the
Klein quartic. Now, we are in a position to relate curves defined
over $\overline{Q} $ via their Belyi-maps and corresponding
dessins to Grothendiecks carthographic groups $\Gamma(2) $,
$\Gamma_0(2) $ and $SL_2(\mathbb{Z}) $. The
dessin gives a permutation representation of the monodromy group and
because the fundamental group of the sphere minus three
points $\pi_1(\mathbb{P}^1 – \\{ 0,1,\infty \\}) =
\langle \sigma_1,\sigma_2,\sigma_3~\mid~\sigma_1 \sigma_2
\sigma_3 = 1 \rangle = \langle \sigma_1,\sigma_2
\rangle $ is the free group op two generators, we see that
any dessin determines a permutation representation of the congruence
subgroup $\Gamma(2) $ (see this
post where we proved that this
group is free). A clean dessin is one for which one type of
vertex has all its valancies (the number of edges in the dessin meeting
the vertex) equal to one or two. (for example, the pre-images of 1 in
the Klein quartic-dessin or the pre-images of 1 in the monsieur Mathieu
example ) The corresponding
permutation $\tau_1 $ then consists of 2-cycles and hence the
monodromy group gives a permutation representation of the free
product $C_{\infty} \ast C_2 =
\Gamma_0(2) $ Finally, a clean dessin is said to be a
quilt dessin if also the other type of vertex has all its valancies
equal to one or three (as in the Klein quartic or Mathieu examples).
Then, the corresponding permutation has order 3 and for these
quilt-dessins the monodromy group gives a permutation representation of
the free product $C_2 \ast C_3 =
PSL_2(\mathbb{Z}) $ Next time we will see how this lead
Grothendieck to his anabelian geometric approach to the absolute Galois
group.

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lulu neverendingbooks

Published July 18, 2005 by lievenlb



Half a year ago, it all started with NeverEndingBooks in which I set out a rather modest goal:

Why NeverEndingBooks? We all complain about exaggerated prices of mathematical books from
certain publishers, poor quality of editing and refereeing offered, as well as far too stringent book-contracts. Rather than lamenting about this, NeverEndingBooks gives itself one year to learn (and report) about the many aspects of the book-production cycle and to explore whether an alternative exists. If at the end of this year we will have produced at least one book this experiment will be considered a success. If,
however, we find out that it is an impossible task, we will explain where it all went wrong and why it is better to stick to an established PublishingHouse and accept its terms.

I assume we did manage to do it after all as you may check by visiting our storefront :
www.lulu.com/neverendingbooks. However, it all turned out to be quite different from what I had in mind half a year ago. So, perhaps it’s time to recap.

Originally, I’d planned to partner-up with the publisher-on-demand LightningSource but in the process they did require a VAT-number. In Belgium, the safest way to get one is to set up a non-profit organization (a VZW as we call
it). But then you have to write down your legal statutes, get them published in the Moniteur Belge (at a hefty price) but what really put me off was that you have to set up a “board of directors” consisting of at least three
people. I don’t mind following a folly but if I have to involve others I usually pass, so I abandoned the whole idea.

Still, I couldn’t help talking about the VAT-problem and at a certain time there was an idea to revive a sleeping VZW (=non-profit organization) of the Belgian Mathematical Society, the MaRC (MAthematical Research Centre), the statutes of which allowed to become a publishing house. But, this wouldn’t involve just two other people but the whole BMS so I decided
to forget all about it and have a short vacation in France together with a few (former)PhD-students.



Given plenty of sun, cheese and whine (not necessarily in that order) sooner or later we had to talk about _the_ problem. For Raf it was the first time he heard about it but when we realized I thought one could easily publish books well under 25 dollars he was immediately interested and insisted we should set up a board of directors and continue with the plan.

The different roles to play in the board were more or less self-evident : I had to be the treasurer (given the fact that I was the only with a secure, though small, income), Geert had to become chairman (being the only one possessing suits), Raf would be secretary (being the only one who could write better Flemish than English) and Jan or Stijn would do PR (as they are the only ones having enough social skills).

So, we went back willing to go through the whole process (at least 3 months) of obtaining a VAT-number.

But then Raf got so interested in the whole idea that he explored other possibilities (I think he was more motivated by the fact that his sister wanted to publish her thesis rather than anything else) and came up with lulu.com.

No legal hassle, no VAT-numbers, nothing required (or so it seemed). Still, before risking his sister’s thesis he wanted to check the service out and as it is a lot easier to take a book lying around rather than write one yourself he took my version 2 and published it at Lulu’s (since then this version is nicknamed Rothko@n).

Although I gave him the permission to do so, it didn’t feel right that people should pay even a small amount for a nicely bound unedited version 2. So, the last month and a half I’ve been editing and partially rewriting version 2 and the two volumes are now available!

Major changes are to the 4 middle chapters. There is now chapter 3 “Etale Technology” which contains all of the etale tricks scattered earlier in two chapters, chapter 4 ‚”Quiver Representations” collects all the
quiver material (again, scattered throughout the previous version). Chapter 5 ‚”Semisimple Representations” now includes recent material such as Raf’s characterization of the smooth locus of Cayley-smooth orders and our (together with Geert) classification of the central singularities, and chapter 6 ‚”Nilpotent Representations” now includes the material on Brauer-Severi varities which was in version 1 but somehow didn’t make it to version 2 before.

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Jacobian update

Published November 13, 2004 by lievenlb

One way to increase the blogshare-value of this site might be to
give readers more of what they want. In fact, there is an excellent
guide for those who really want to increase traffic on their site
called 26
Steps to 15k a Day. A somewhat sobering suggestion is rule S :

“Think about what people want. They
aren't coming to your site to view “your content”,
they are coming to your site looking for “their
content”.”

But how do we know what
people want? Well, by paying attention to Google-referrals according
to rule U :

“The search engines will
tell you exactly what they want to be fed – listen closely, there is
gold in referral logs, it's just a matter of panning for
it.”

And what do these Google-referrals
show over the last couple of days? Well, here are the top recent
key-words given to Google to get here :

13 :
carolyn dean jacobian conjecture
11 : carolyn dean jacobian

9 : brauer severi varieties
7 : latexrender

7 : brauer severi
7 : spinor bundles
7 : ingalls
azumaya
6 : [Unparseable or potentially dangerous latex
formula Error 6 ]
6 : jacobian conjecture carolyn dean

See a pattern? People love to hear right now about
the solution of the Jacobian conjecture in the plane by Carolyn Dean.
Fortunately, there are a couple of things more I can say about this
and it may take a while before you know why there is a photo of Tracy
Chapman next to this post…

First, it seems I only got
part of the Melvin Hochster
email. Here is the final part I was unaware of (thanks to not even wrong)

Earlier papers established the following: if
there is
a counterexample, the leading forms of $f$ and $g$
may
be assumed to have the form $(x^a y^b)^J$ and $(x^a
y^b)^K$,
where $a$ and $b$ are relatively prime and neither
$J$
nor $K$ divides the other (Abhyankar, 1977). It is known
that
$a$ and $b$ cannot both be $1$ (Lang, 1991) and that one
may
assume that $C[f,g]$ does not contain a degree one
polynomial
in $x, y$ (Formanek, 1994).

Let $D_x$ and $D_y$ indicate partial differentiation with respect

to $x$ and $y$, respectively. A difficult result of Bass (1989)

asserts that if $D$ is a non-zero operator that is a polynomial

over $C$ in $x D_x$ and $y D_y$, $G$ is in $C[x,y]$ and $D(G)$

is in $C[f,g]$, then $G$ is in $C[f,g]$.

The proof
proceeds by starting with $f$ and $g$ that give
a
counterexample, and recursively constructing sequences of
elements and derivations with remarkable, intricate and
surprising relationships. Ultimately, a contradiction is
obtained by studying a sequence of positive integers associated
with the degrees of the elements constructed. One delicate
argument shows that the sequence is bounded. Another delicate
argument shows that it is not. Assuming the results described
above, the proof, while complicated, is remarkably self-contained
and can be understood with minimal background in algebra.

  • Mel Hochster

Speaking about the Jacobian
conjecture-post at not even wrong and
the discussion in the comments to it : there were a few instances I
really wanted to join in but I'll do it here. To begin, I was a
bit surprised of the implicit attack in the post

Dean hasn't published any papers in almost 15 years and is
nominally a lecturer in mathematics education at Michigan.

But this was immediately addressed and retracted in
the comments :

Just curious. What exactly did
you mean by “nominally a lecturer”?
Posted by mm
at November 10, 2004 10:54 PM

I don't know
anything about Carolyn Dean personally, just that one place on the
Michigan web-site refers to her as a “lecturer”, another
as a “visiting lecturer”. As I'm quite well aware from
personal experience, these kinds of titles can refer to all sorts of
different kinds of actual positions. So the title doesn't tell you
much, which is what I was awkwardly expressing.
Posted by Peter
at November 10, 2004 11:05 PM

Well, I know a few things
about Carolyn Dean personally, the most relevant being that she is a
very careful mathematician. I met her a while back (fall of 1985) at
UCSD where she was finishing (or had finished) her Ph.D. If Lance
Small's description of me would have been more reassuring, we
might even have ended up sharing an apartment (quod non). Instead I
ended up with Claudio
Procesi… Anyway, it was a very enjoyable month with a group
of young starting mathematicians and I fondly remember some
dinner-parties we organized. The last news I heard about Carolyn was
10 to 15 years ago in Oberwolfach when it was rumoured that she had
solved the Jacobian conjecture in the plane… As far as I recall,
the method sketched by Hochster in his email was also the one back
then. Unfortunately, at the time she still didn't have all pieces
in place and a gap was found (was it by Toby Stafford? or was it
Hochster?, I forgot). Anyway, she promptly acknowledged that there was
a gap.
At the time I was dubious about the approach (mostly
because I was secretly trying to solve it myself) but today my gut
feeling is that she really did solve it. In recent years there have
been significant advances in polynomial automorphisms (in particular
the tame-wild problem) and in the study of the Hilbert scheme of
points in the plane (which I always thought might lead to a proof) so
perhaps some of these recent results did give Carolyn clues to finish
off her old approach? I haven't seen one letter of the proof so
I'm merely speculating here. Anyway, Hochster's assurance that
the proof is correct is good enough for me right now.
Another
discussion in the NotEvenWrong-comments was on the issue that several
old problems were recently solved by people who devoted themselves for
several years solely to that problem and didn't join the parade of
dedicated follower of fashion-mathematicians.

It is remarkable that the last decade has seen great progress in
math (Wiles proving Fermat's Last Theorem, Perelman proving the
Poincare Conjecture, now Dean the Jacobian Conjecture), all achieved
by people willing to spend 7 years or more focusing on a single
problem. That's not the way academic research is generally
structured, if you want grants, etc. you should be working on much
shorter term projects. It's also remarkable that two out of three
of these people didn't have a regular tenured position.

I think particle theory should learn from this. If
some of the smarter people in the field would actually spend 7 years
concentrating on one problem, the field might actually go somewhere
instead of being dead in the water
Posted by Peter at November
13, 2004 08:56 AM

Here we come close to a major problem of
today's mathematics. I have the feeling that far too few
mathematicians dedicate themselves to problems in which they have a
personal interest, independent of what the rest of the world might
think about these problems. Far too many resort to doing trendy,
technical mathematics merely because it is approved by so called
'better' mathematicians. Mind you, I admit that I did fall in
that trap myself several times but lately I feel quite relieved to be
doing just the things I like to do no matter what the rest may think
about it. Here is a little bit of advice to some colleagues : get
yourself an iPod and take
some time to listen to songs like this one :

Don't be tempted by the shiny apple
Don't you eat
of a bitter fruit
Hunger only for a taste of justice

Hunger only for a world of truth
'Cause all that you have
is your soul

from Tracy Chapman's All
that you have is your soul

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reading backlog

Published November 8, 2004 by lievenlb

One of the things I like most about returning from a vacation is to
have an enormous pile of fresh reading : a week's worth of
newspapers, some regular mail and much more email (three quarters junk).
Also before getting into bed after the ride I like to browse through the
arXiv in search for interesting
papers.
This time, the major surprise of my initial survey came
from the newspapers. No, not Bush again, _that_ news was headline
even in France. On the other hand, I didn't hear a word about Theo Van
Gogh being shot and stabbed to death in Amsterdam. I'll come
back to this later.
I'd rather mention the two papers that
somehow stood out during my scan of this week on the arXiv. The first is
Framed quiver moduli,
cohomology, and quantum groups by Markus
Reineke. By the deframing trick, a framed quiver moduli problem is
reduced to an ordinary quiver moduli problem for a dimension vector for
which one of the entries is equal to one, hence in particular, an
indivisible dimension vector. Such quiver problems are far easier to
handle than the divisible ones where everything can at best be reduced
to the classical problem of classifying tuples of $n \\times n$ matrices
up to simultaneous conjugation. Markus deals with the case when the
quiver has no oriented cycles. An important examples of a framed moduli
quiver problem _with_ oriented cycles is the study of
Brauer-Severi varieties of smooth orders. Significant progress on the
description of the fibers in this case is achieved by Raf Bocklandt,
Stijn Symens and Geert Van de Weyer and will (hopefully) be posted soon.

The second paper is Moduli schemes of rank
one Azumaya modules by Norbert Hoffmann and Urich Stuhler which
brings back longforgotten memories of my Ph.D. thesis, 21 years
ago…

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hyper-resolutions

Published September 30, 2004 by lievenlb

[Last time][1] we saw that for $A$ a smooth order with center $R$ the
Brauer-Severi variety $X_A$ is a smooth variety and we have a projective
morphism $X_A \rightarrow \mathbf{max}~R$ This situation is
very similar to that of a desingularization $~X \rightarrow
\mathbf{max}~R$ of the (possibly singular) variety $~\mathbf{max}~R$.
The top variety $~X$ is a smooth variety and there is a Zariski open
subset of $~\mathbf{max}~R$ where the fibers of this map consist of just
one point, or in more bombastic language a $~\mathbb{P}^0$. The only
difference in the case of the Brauer-Severi fibration is that we have a
Zariski open subset of $~\mathbf{max}~R$ (the Azumaya locus of A) where
the fibers of the fibration are isomorphic to $~\mathbb{P}^{n-1}$. In
this way one might view the Brauer-Severi fibration of a smooth order as
a non-commutative or hyper-desingularization of the central variety.
This might provide a way to attack the old problem of construction
desingularizations of quiver-quotients. If $~Q$ is a quiver and $\alpha$
is an indivisible dimension vector (that is, the component dimensions
are coprime) then it is well known (a result due to [Alastair King][2])
that for a generic stability structure $\theta$ the moduli space
$~M^{\theta}(Q,\alpha)$ classifying $\theta$-semistable
$\alpha$-dimensional representations will be a smooth variety (as all
$\theta$-semistables are actually $\theta$-stable) and the fibration
$~M^{\theta}(Q,\alpha) \rightarrow \mathbf{iss}_{\alpha}~Q$ is a
desingularization of the quotient-variety $~\mathbf{iss}_{\alpha}~Q$
classifying isomorphism classes of $\alpha$-dimensional semi-simple
representations. However, if $\alpha$ is not indivisible nobody has
the faintest clue as to how to construct a natural desingularization of
$~\mathbf{iss}_{\alpha}~Q$. Still, we have a perfectly reasonable
hyper-desingularization $~X_{A(Q,\alpha)} \rightarrow
\mathbf{iss}_{\alpha}~Q$ where $~A(Q,\alpha)$ is the corresponding
quiver order, the generic fibers of which are all projective spaces in
case $\alpha$ is the dimension vector of a simple representation of
$~Q$. I conjecture (meaning : I hope) that this Brauer-Severi fibration
contains already a lot of information on a genuine desingularization of
$~\mathbf{iss}_{\alpha}~Q$. One obvious test for this seemingly
crazy conjecture is to study the flat locus of the Brauer-Severi
fibration. If it would contain info about desingularizations one would
expect that the fibration can never be flat in a central singularity! In
other words, we would like that the flat locus of the fibration is
contained in the smooth central locus. This is indeed the case and is a
more or less straightforward application of the proof (due to [Geert Van
de Weyer][3]) of the Popov-conjecture for quiver-quotients (see for
example his Ph.D. thesis [Nullcones of quiver representations][4]).
However, it is in general not true that the flat-locus and central
smooth locus coincide. Sometimes this is because the Brauer-Severi
scheme is a blow-up of the Brauer-Severi of a nicer order. The following
example was worked out together with [Colin Ingalls][5] : Consider the
order $~A = \begin{bmatrix} C[x,y] & C[x,y] \\ (x,y) & C[x,y]
\end{bmatrix}$ which is the quiver order of the quiver setting
$~(Q,\alpha)$ $\xymatrix{\vtx{1} \ar@/^2ex/[rr] \ar@/^1ex/[rr]
& & \vtx{1} \ar@/^2ex/[ll]} $ then the Brauer-Severi fibration
$~X_A \rightarrow \mathbf{iss}_{\alpha}~Q$ is flat everywhere except
over the zero representation where the fiber is $~\mathbb{P}^1 \times
\mathbb{P}^2$. On the other hand, for the order $~B =
\begin{bmatrix} C[x,y] & C[x,y] \\ C[x,y] & C[x,y] \end{bmatrix}$
the Brauer-Severi fibration is flat and $~X_B \simeq \mathbb{A}^2 \times
\mathbb{P}^1$. It turns out that $~X_A$ is a blow-up of $~X_B$ at a
point in the fiber over the zero-representation.

[1]: http://www.neverendingbooks.org/index.php?p=342
[2]: http://www.maths.bath.ac.uk/~masadk/
[3]: http://www.win.ua.ac.be/~gvdwey/
[4]: http://www.win.ua.ac.be/~gvdwey/papers/thesis.pdf
[5]: http://kappa.math.unb.ca/~colin/

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smooth Brauer-Severis

Published September 28, 2004 by lievenlb

Around the
same time Michel Van den Bergh introduced his Brauer-Severi schemes,
[Claudio Procesi][1] (extending earlier work of [Bill Schelter][2])
introduced smooth orders as those orders $A$ in a central simple algebra
$\Sigma$ (of dimension $n^2$) such that their representation variety
$\mathbf{trep}_n~A$ is a smooth variety. Many interesting orders are smooth
: hereditary orders, trace rings of generic matrices and more generally
size n approximations of formally smooth algebras (that is,
non-commutative manifolds). As in the commutative case, every order has
a Zariski open subset where it is a smooth order. The relevance of
this notion to the study of Brauer-Severi varieties is that $X_A$ is a
smooth variety whenever $A$ is a smooth order. Indeed, the Brauer-Severi
scheme was the orbit space of the principal $GL_n$-fibration on the
Brauer-stable representations (see [last time][3]) which form a Zariski
open subset of the smooth variety $\mathbf{trep}_n~A \times k^n$. In fact,
in most cases the reverse implication will also hold, that is, if $X_A$
is smooth then usually A is a smooth order. However, for low n,
there are some counterexamples. Consider the so called quantum plane
$A_q=k_q[x,y]~:~yx=qxy$ with $~q$ an $n$-th root of unity then one
can easily prove (using the fact that the smooth order locus of $A_q$ is
everything but the origin in the central variety $~\mathbb{A}^2$) that
the singularities of the Brauer-Severi scheme $X_A$ are the orbits
corresponding to those nilpotent representations $~\phi : A \rightarrow
M_n(k)$ which are at the same time singular points in $\mathbf{trep}_n~A$
and have a cyclic vector. As there are singular points among the
nilpotent representations, the Brauer-Severi scheme will also be
singular except perhaps for small values of $n$. For example, if
$~n=2$ the defining relation is $~xy+yx=0$ and any trace preserving
representation has a matrix-description $~x \rightarrow
\begin{bmatrix} a & b \\ c & -a \end{bmatrix}~y \rightarrow
\begin{bmatrix} d & e \\ f & -d \end{bmatrix}$ such that
$~2ad+bf+ec = 0$. That is, $~\mathbf{trep}_2~A = \mathbb{V}(2ad+bf+ec)
\subset \mathbb{A}^6$ which is an hypersurface with a unique
singular point (the origin). As this point corresponds to the
zero-representation (which does not have a cyclic vector) the
Brauer-Severi scheme will be smooth in this case. [Colin
Ingalls][4] extended this calculation to show that the Brauer-Severi
scheme is equally smooth when $~n=3$ but has a unique (!) singular point
when $~n=4$. So probably all Brauer-Severi schemes for $n \geq 4$ are
indeed singular. I conjecture that this is a general feature for
Brauer-Severi schemes of families (depending on the p.i.-degree $n$) of
non-smooth orders.

[1]: http://venere.mat.uniroma1.it/people/procesi/
[2]: http://www.fact-index.com/b/bi/bill_schelter.html
[3]: http://www.neverendingbooks.org/index.php?p=341
[4]: http://kappa.math.unb.ca/~colin/

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Brauer-Severi varieties

Published September 27, 2004 by lievenlb

![][1]
Classical Brauer-Severi varieties can be described either as twisted
forms of projective space (Severi\’s way) or as varieties containing
splitting information about central simple algebras (Brauer\’s way). If
$K$ is a field with separable closure $\overline{K}$, the first approach
asks for projective varieties $X$ defined over $K$ such that over the
separable closure $X(\overline{K}) \simeq
\mathbb{P}^{n-1}_{\overline{K}}$ they are just projective space. In
the second approach let $\Sigma$ be a central simple $K$-algebra and
define a variety $X_{\Sigma}$ whose points over a field extension $L$
are precisely the left ideals of $\Sigma \otimes_K L$ of dimension $n$.
This variety is defined over $K$ and is a closed subvariety of the
Grassmannian $Gr(n,n^2)$. In the special case that $\Sigma = M_n(K)$ one
can use the matrix-idempotents to show that the left ideals of dimension
$n$ correspond to the points of $\mathbb{P}^{n-1}_K$. As for any central
simple $K$-algebra $\Sigma$ we have that $\Sigma \otimes_K \overline{K}
\simeq M_n(\overline{K})$ it follows that the varieties $X_{\Sigma}$ are
among those of the first approach. In fact, there is a natural bijection
between those of the first approach (twisted forms) and of the second as
both are classified by the Galois cohomology pointed set
$H^1(Gal(\overline{K}/K),PGL_n(\overline{K}))$ because
$PGL_n(\overline{K})$ is the automorphism group of
$\mathbb{P}^{n-1}_{\overline{K}}$ as well as of $M_n(\overline{K})$. The
ringtheoretic relevance of the Brauer-Severi variety $X_{\Sigma}$ is
that for any field extension $L$ it has $L$-rational points if and only
if $L$ is a _splitting field_ for $\Sigma$, that is, $\Sigma \otimes_K L
\simeq M_n(\Sigma)$. To give one concrete example, If $\Sigma$ is the
quaternion-algebra $(a,b)_K$, then the Brauer-Severi variety is a conic
$X_{\Sigma} = \mathbb{V}(x_0^2-ax_1^2-bx_2^2) \subset \mathbb{P}^2_K$
Whenever one has something working for central simple algebras, one can
_sheafify_ the construction to Azumaya algebras. For if $A$ is an
Azumaya algebra with center $R$ then for every maximal ideal
$\mathfrak{m}$ of $R$, the quotient $A/\mathfrak{m}A$ is a central
simple $R/\mathfrak{m}$-algebra. This was noted by the
sheafification-guru [Alexander Grothendieck][2] and he extended the
notion to Brauer-Severi schemes of Azumaya algebras which are projective
bundles $X_A \rightarrow \mathbf{max}~R$ all of which fibers are
projective spaces (in case $R$ is an affine algebra over an
algebraically closed field). But the real fun started when [Mike
Artin][3] and [David Mumford][4] extended the construction to suitably
_ramified_ algebras. In good cases one has that the Brauer-Severi
fibration is flat with fibers over ramified points certain degenerations
of projective space. For example in the case considered by Artin and
Mumford of suitably ramified orders in quaternion algebras, the smooth
conics over Azumaya points degenerate to a pair of lines over ramified
points. A major application of their construction were examples of
unirational non-rational varieties. To date still one of the nicest
applications of non-commutative algebra to more mainstream mathematics.
The final step in generalizing Brauer-Severi fibrations to arbitrary
orders was achieved by [Michel Van den Bergh][5] in 1986. Let $R$ be an
affine algebra over an algebraically closed field (say of characteristic
zero) $k$ and let $A$ be an $R$-order is a central simple algebra
$\Sigma$ of dimension $n^2$. Let $\mathbf{trep}_n~A$ be teh affine variety
of _trace preserving_ $n$-dimensional representations, then there is a
natural action of $GL_n$ on this variety by basechange (conjugation).
Moreover, $GL_n$ acts by left multiplication on column vectors $k^n$.
One then considers the open subset in $\mathbf{trep}_n~A \times k^n$
consisting of _Brauer-Stable representations_, that is those pairs
$(\phi,v)$ such that $\phi(A).v = k^n$ on which $GL_n$ acts freely. The
corresponding orbit space is then called the Brauer-Severio scheme $X_A$
of $A$ and there is a fibration $X_A \rightarrow \mathbf{max}~R$ again
having as fibers projective spaces over Azumaya points but this time the
fibration is allowed to be far from flat in general. Two months ago I
outlined in Warwick an idea to apply this Brauer-Severi scheme to get a
hold on desingularizations of quiver quotient singularities. More on
this next time.

[1]: http://www.neverendingbooks.org/DATA/brauer.jpg
[2]: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.html
[3]: http://www.cirs-tm.org/researchers/researchers.php?id=235
[4]: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Mumford.html
[5]: http://alpha.luc.ac.be/Research/Algebra/Members/michel_id.html

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Galois and the Brauer group

Published February 9, 2004 by lievenlb

Last time we have seen that in order to classify all
non-commutative $l$-points one needs to control the finite
dimensional simple algebras having as their center a finite
dimensional field-extension of $l$. We have seen that the equivalence
classes of simple algebras with the same center $L$ form an Abelian
group, the
Brauer group. The calculation of Brauer groups
is best done using
Galois-cohomology. As an aside :
Evariste Galois was one of the more tragic figures in the history of
mathematics, he died at the age of 20 as a result of a duel. There is
a whole site the Evariste Galois archive dedicated to his
work.

But let us return to a simple algebra $T$ over the
field $L$ which we have seen to be of the form $M(k,S)$, full
matrices over a division algebra $S$. We know that the dimension of
$S$ over $L$ is a square, say $n^2$, and it can be shown that all
maximal commutative subfields of $S$ have dimension n over $L$.
In this way one can view a simple algebra as a bag containing all
sorts of degree n extensions of its center. All these maximal
subfields are also splitting fields for $S$, meaning that
if you tensor $S$ with one of them, say $M$, one obtains full nxn
matrices $M(n,M)$. Among this collection there is at least one
separable field but for a long time it was an open question
whether the collection of all maximal commutative subfields also
contains a Galois-extension of $L$. If this is the case, then
one could describe the division algebra $S$ as a crossed
product. It was known for some time that there is always a simple
algebra $S’$ equivalent to $S$ which is a crossed product (usually
corresponding to a different number n’), that is, all elements of
the Brauer group can be represented by crossed products. It came as a
surprise when S.A. Amitsur in 1972 came up with examples of
non-crossed product division algebras, that is, division algebras $D$
such that none of its maximal commutative subfields is a Galois
extension of the center. His examples were generic
division algebras $D(n)$. To define $D(n)$ take two generic
nxn matrices, that is, nxn matrices A and B such that all its
entries are algebraically independent over $L$ and consider the
$L$-subalgebra generated by A and B in the full nxn matrixring over the
field $F$ generated by all entries of A and B. Somewhat surprisingly,
one can show that this subalgebra is a domain and inverting all its
central elements (which, again, is somewhat of a surprise that
there are lots of them apart from elements of $L$, the so called
central polynomials) one obtains the division algebra $D(n)$ with
center $F(n)$ which has trancendence degree n^2 1 over $L$. By the
way, it is still unknown (apart from some low n cases) whether $F(n)$
is purely trancendental over $L$. Now, utilising the generic
nature of $D(n)$, Amitsur was able to prove that when $L=Q$, the
field of rational numbers, $D(n)$ cannot be a crossed product unless
$n=2^s p_1…p_k$ with the p_i prime numbers and s at most 2. So, for
example $D(8)$ is not a crossed product.

One can then
ask whether any division algebra $S$, of dimension n^2 over $L$, is a
crossed whenever n is squarefree. Even teh simplest case, when n is a
prime number is not known unless p=2 or 3. This shows how little we do
know about finite dimensional division algebras : nobody knows
whether a division algebra of dimension 25 contains a maximal
cyclic subfield (the main problem in deciding this type of
problems is that we know so few methods to construct division
algebras; either they are constructed quite explicitly as a crossed
product or otherwise they are constructed by some generic construction
but then it is very hard to make explicit calculations with
them).

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