# Tag: Brauer

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…

[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
[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/

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/

![][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

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).