non-commutative – Page 12 – neverendingbooks

hyper-resolutions

Published September 30, 2004 by lieven

[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 $X_{A} \to \max R$ This situation is
very similar to that of a desingularization $X \to \max R$ of the (possibly singular) variety $\max R$ .
The top variety $X$ is a smooth variety and there is a Zariski open
subset of $\max R$ where the fibers of this map consist of just
one point, or in more bombastic language a $P^{0}$ . The only
difference in the case of the Brauer-Severi fibration is that we have a
Zariski open subset of $\max R$ (the Azumaya locus of A) where
the fibers of the fibration are isomorphic to $P^{n - 1}$ . 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 $α$
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 $θ$ the moduli space
$M^{θ} (Q, α)$ classifying $θ$ -semistable
$α$ -dimensional representations will be a smooth variety (as all
$θ$ -semistables are actually $θ$ -stable) and the fibration
$M^{θ} (Q, α) \to {iss}_{α} Q$ is a
desingularization of the quotient-variety ${iss}_{α} Q$
classifying isomorphism classes of $α$ -dimensional semi-simple
representations. However, if $α$ is not indivisible nobody has
the faintest clue as to how to construct a natural desingularization of
${iss}_{α} Q$ . Still, we have a perfectly reasonable
hyper-desingularization $X_{A (Q, α)} \to {iss}_{α} Q$ where $A (Q, α)$ is the corresponding
quiver order, the generic fibers of which are all projective spaces in
case $α$ 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
${iss}_{α} Q$ . 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 $A = [\begin{matrix} C [x, y] & C [x, y] \\ (x, y) & C [x, y] \end{matrix}]$ which is the quiver order of the quiver setting
$(Q, α)$ $Misplaced &$ then the Brauer-Severi fibration
$X_{A} \to {iss}_{α} Q$ is flat everywhere except
over the zero representation where the fiber is $P^{1} \times P^{2}$ . On the other hand, for the order $B = [\begin{matrix} C [x, y] & C [x, y] \\ C [x, y] & C [x, y] \end{matrix}]$
the Brauer-Severi fibration is flat and $X_{B} ≃ A^{2} \times P^{1}$ . It turns out that $X_{A}$ is a blow-up of $X_{B}$ at a
point in the fiber over the zero-representation.

[1]: https://lievenlb.local/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/

smooth Brauer-Severis

Published September 28, 2004 by lieven

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
$Σ$ (of dimension $n^{2}$ ) such that their representation variety
${trep}_{n} A$ 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 $G L_{n}$ -fibration on the
Brauer-stable representations (see [last time][3]) which form a Zariski
open subset of the smooth variety ${trep}_{n} A \times k^{n}$ . 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
$A_{q} = k_{q} [x, y] : y x = q x y$ 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 $A^{2}$ ) that
the singularities of the Brauer-Severi scheme $X_{A}$ are the orbits
corresponding to those nilpotent representations $ϕ : A \to M_{n} (k)$ which are at the same time singular points in ${trep}_{n} A$
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 $x y + y x = 0$ and any trace preserving
representation has a matrix-description $x \to [\begin{matrix} a & b \\ c & - a \end{matrix}] y \to [\begin{matrix} d & e \\ f & - d \end{matrix}]$ such that
$2 a d + b f + e c = 0$ . That is, ${trep}_{2} A = V (2 a d + b f + e c) \subset A^{6}$ 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]: https://lievenlb.local/index.php?p=341
[4]: http://kappa.math.unb.ca/~colin/

Brauer-Severi varieties

Published September 27, 2004 by lieven

![][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 $\overset{―}{K}$ , the first approach
asks for projective varieties $X$ defined over $K$ such that over the
separable closure $X (\overset{―}{K}) ≃ P_{\overset{―}{K}}^{n - 1}$ they are just projective space. In
the second approach let $Σ$ be a central simple $K$ -algebra and
define a variety $X_{Σ}$ whose points over a field extension $L$
are precisely the left ideals of $Σ \otimes_{K} L$ of dimension $n$ .
This variety is defined over $K$ and is a closed subvariety of the
Grassmannian $G r (n, n^{2})$ . In the special case that $Σ = M_{n} (K)$ one
can use the matrix-idempotents to show that the left ideals of dimension
$n$ correspond to the points of $P_{K}^{n - 1}$ . As for any central
simple $K$ -algebra $Σ$ we have that $Σ \otimes_{K} \overset{―}{K} ≃ M_{n} (\overset{―}{K})$ it follows that the varieties $X_{Σ}$ 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
$H^{1} (G a l (\overset{―}{K} / K), P G L_{n} (\overset{―}{K}))$ because
$P G L_{n} (\overset{―}{K})$ is the automorphism group of
$P_{\overset{―}{K}}^{n - 1}$ as well as of $M_{n} (\overset{―}{K})$ . The
ringtheoretic relevance of the Brauer-Severi variety $X_{Σ}$ is
that for any field extension $L$ it has $L$ -rational points if and only
if $L$ is a _splitting field_ for $Σ$ , that is, $Σ \otimes_{K} L ≃ M_{n} (Σ)$ . To give one concrete example, If $Σ$ is the
quaternion-algebra $(a, b)_{K}$ , then the Brauer-Severi variety is a conic
$X_{Σ} = V (x_{0}^{2} - a x_{1}^{2} - b x_{2}^{2}) \subset P_{K}^{2}$
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
$m$ of $R$ , the quotient $A / m A$ is a central
simple $R / m$ -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 $X_{A} \to \max R$ 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
$Σ$ of dimension $n^{2}$ . Let ${trep}_{n} A$ be teh affine variety
of _trace preserving_ $n$ -dimensional representations, then there is a
natural action of $G L_{n}$ on this variety by basechange (conjugation).
Moreover, $G L_{n}$ acts by left multiplication on column vectors $k^{n}$ .
One then considers the open subset in ${trep}_{n} A \times k^{n}$
consisting of _Brauer-Stable representations_, that is those pairs
$(ϕ, v)$ such that $ϕ (A) . v = k^{n}$ on which $G L_{n}$ acts freely. The
corresponding orbit space is then called the Brauer-Severio scheme $X_{A}$
of $A$ and there is a fibration $X_{A} \to \max R$ 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]: https://lievenlb.local/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

Tag: non-commutative

hyper-resolutions

smooth Brauer-Severis

Brauer-Severi varieties