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Tag: Azumaya

hyper-resolutions

[Last time][1] we saw that for A a smooth order with center R the
Brauer-Severi variety XA is a smooth variety and we have a projective
morphism XAmax R This situation is
very similar to that of a desingularization  Xmax 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  P0. 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  Pn1. 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
 Mθ(Q,α) classifying θ-semistable
α-dimensional representations will be a smooth variety (as all
θ-semistables are actually θ-stable) and the fibration
 Mθ(Q,α)issα Q is a
desingularization of the quotient-variety  issα 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
 issα Q. Still, we have a perfectly reasonable
hyper-desingularization  XA(Q,α)issα 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
 issα 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=[C[x,y]C[x,y](x,y)C[x,y]] which is the quiver order of the quiver setting
 (Q,α) Misplaced & then the Brauer-Severi fibration
 XAissα Q is flat everywhere except
over the zero representation where the fiber is  P1×P2. On the other hand, for the order  B=[C[x,y]C[x,y]C[x,y]C[x,y]]
the Brauer-Severi fibration is flat and  XBA2×P1. It turns out that  XA is a blow-up of  XB 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/

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Brauer-Severi varieties

![][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 K, the first approach
asks for projective varieties X defined over K such that over the
separable closure X(K)PKn1 they are just projective space. In
the second approach let Σ be a central simple K-algebra and
define a variety XΣ whose points over a field extension L
are precisely the left ideals of ΣKL of dimension n.
This variety is defined over K and is a closed subvariety of the
Grassmannian Gr(n,n2). In the special case that Σ=Mn(K) one
can use the matrix-idempotents to show that the left ideals of dimension
n correspond to the points of PKn1. As for any central
simple K-algebra Σ we have that ΣKKMn(K) it follows that the varieties XΣ 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
H1(Gal(K/K),PGLn(K)) because
PGLn(K) is the automorphism group of
PKn1 as well as of Mn(K). The
ringtheoretic relevance of the Brauer-Severi variety XΣ 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, ΣKLMn(Σ). To give one concrete example, If Σ is the
quaternion-algebra (a,b)K, then the Brauer-Severi variety is a conic
XΣ=V(x02ax12bx22)PK2
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/mA 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 XAmax 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 n2. Let trepn A be teh affine variety
of _trace preserving_ n-dimensional representations, then there is a
natural action of GLn on this variety by basechange (conjugation).
Moreover, GLn acts by left multiplication on column vectors kn.
One then considers the open subset in trepn A×kn
consisting of _Brauer-Stable representations_, that is those pairs
(ϕ,v) such that ϕ(A).v=kn on which GLn acts freely. The
corresponding orbit space is then called the Brauer-Severio scheme XA
of A and there is a fibration XAmax 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

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the Azumaya locus does determine the order

Clearly
this cannot be correct for consider for nN the order

An=[C[x]C[x](xn)C[x]]

For mn the orders An
and Am have isomorphic Azumaya locus, but are not isomorphic as
orders. Still, the statement in the heading is _morally_ what Nikolaus
Vonessen
and Zinovy
Reichstein
are proving in their paper Polynomial identity
rings as rings of functions
. So I better clarify what they do claim
precisely.

Let A be a _Cayley-Hamilton order_, that is, a
prime affine C-algebra, finite as a module over its center
and satisfying all trace relations holding in Mn(C). If A
is generated by m elements, then its _representation variety_
repn A has as points the m-tuples of n×n matrices

(X1,,Xm)Mn(C)Mn(C)

which satisfy all the defining relations of
A. repn A is an affine variety with a GLn-action
(induced by simultaneous conjugation in m-tuples of matrices) and has
as a Zariski open subset the tuples (X1,,Xm)repn A having the property that they generate the whole
matrix-algebra Mn(C). This open subset is called the
Azumaya locus of A and denoted by azun A.

One can also define the _generic Azumaya locus_ as being the
Zariski open subset of Mn(C)Mn(C) consisting of those tuples which generate
Mn(C) and call this subset Azun. In fact, one
can show that Azun is the Azumaya locus of a particular
order namely the trace ring of m generic n×n matrices.

What Nikolaus and Zinovy prove is that for an order A the Azumaya
locus azun A is an irreducible subvariety of
Azun and that the embedding

azun AAzun

determines A itself! If you have
worked a bit with orders this result is strange at first until you
recognize it as being essentially a consequence of Bill Schelter's
catenarity result for affine p.i.-algebras.

On the positive
side it shows that the study of orders is roughly equivalent to that of
the study of irreducible GLn-stable subvarieties of Azun.
On the negative side, it shows that the GLn-structure of
Azun is horribly complicated. For example, it is still
unknown in general whether the quotient-variety (which is here also the
orbit space) Azun/GLn is a rational variety.

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