the Azumaya locus does determine the order

Clearly this cannot be correct for consider for the order

For the orders and 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 be a Cayley-Hamilton order, that is, a prime affine -algebra, finite as a module over its center and satisfying all trace relations holding in . If is generated by elements, then its _representation variety has as points the m-tuples of matrices

tex \in Mn(\mathbb{C}) \oplus \hdots \oplus Mn(\mathbb{C})[/tex]

which satisfy all the defining relations of A. is an affine variety with a -action (induced by simultaneous conjugation in m-tuples of matrices) and has as a Zariski open subset the tuples tex \in \mathbf{rep}n~A[/tex] having the property that they generate the whole matrix-algebra . This open subset is called the Azumaya locus of A and denoted by .

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

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

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 -stable subvarieties of . On the negative side, it shows that the -structure of is horribly complicated. For example, it is still unknown in general whether the quotient-variety (which is here also the orbit space) is a rational variety.