# the Azumaya locus does determine the order

Clearly
this cannot be correct for consider for $n \in \mathbb{N}$ the order

$A_n = \begin{bmatrix} \mathbb{C}[x] & \mathbb{C}[x] \\ (x^n) & \mathbb{C}[x] \end{bmatrix}$

For $m \not= n$ the orders $A_n$
and $A_m$ 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 $\mathbb{C}$-algebra, finite as a module over its center
and satisfying all trace relations holding in $M_n(\mathbb{C})$. If $A$
is generated by $m$ elements, then its _representation variety_
$\mathbf{rep}_n~A$ has as points the m-tuples of $n \times n$ matrices

$(X_1,\ldots,X_m) \in M_n(\mathbb{C}) \oplus \ldots \oplus M_n(\mathbb{C})$

which satisfy all the defining relations of
A. $\mathbf{rep}_n~A$ is an affine variety with a $GL_n$-action
(induced by simultaneous conjugation in m-tuples of matrices) and has
as a Zariski open subset the tuples $(X_1,\ldots,X_m) \in \mathbf{rep}_n~A$ having the property that they generate the whole
matrix-algebra $M_n(\mathbb{C})$. This open subset is called the
Azumaya locus of A and denoted by $\mathbf{azu}_n~A$.

One can also define the _generic Azumaya locus_ as being the
Zariski open subset of $M_n(\mathbb{C}) \oplus \ldots \oplus M_n(\mathbb{C})$ consisting of those tuples which generate
$M_n(\mathbb{C})$ and call this subset $\mathbf{Azu}_n$. In fact, one
can show that $\mathbf{Azu}_n$ is the Azumaya locus of a particular
order namely the trace ring of m generic $n \times n$ matrices.

What Nikolaus and Zinovy prove is that for an order A the Azumaya
locus $\mathbf{azu}_n~A$ is an irreducible subvariety of
$\mathbf{Azu}_n$ and that the embedding

$\mathbf{azu}_n~A \subset \mathbf{Azu}_n$

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 $GL_n$-stable subvarieties of $\mathbf{Azu}_n$.
On the negative side, it shows that the $GL_n$-structure of
$\mathbf{Azu}_n$ is horribly complicated. For example, it is still
unknown in general whether the quotient-variety (which is here also the
orbit space) $\mathbf{Azu}_n / GL_n$ is a rational variety.

### Similar Posts:

This site uses Akismet to reduce spam. Learn how your comment data is processed.