Bounding the minimal number of generators of an Azumaya algebra

Abstract

A paper of U. First & Z. Reichstein proves that if $R$ is a commutative ring of dimension $d$, then any Azumaya algebra $A$ over $R$ can be generated as an algebra by $d+2$ elements, by constructing such a generating set, but they do not prove that this number of generators is required, or even that for an arbitrarily large $r$ that there exists an Azumaya algebra requiring $r$ generators. In this paper, for any given fixed $n\ge 2$, we produce examples of a base ring $R$ of dimension $d$ and an Azumaya algebra of degree $n$ over $R$ that requires $r(d,n)=\lfloor \frac{d}{2n-2}\rfloor +2$ generators. While $r(d,n)<d+2$ in general, we at least show that there is no uniform upper bound on the number of generators required for Azumaya algebras. The method of proof is to consider certain varieties $B_n^r$ that are universal varieties for degree-$n$ Azumaya algebras equipped with a set of $r$ generators, and specifically we show that a natural map on Chow group $C{H}_{PG{L}_n}^{(r-1)(n-1)}\to C{H}^{(r-1)(n-1)}(B_n^r)$ fails to be injective, which is to say that the map fails to be injective in the first dimension in which it possibly could fail. This implies that for a sufficiently generic rank-$n$ Azumaya algebra, there is a characteristic class obstruction to generation by $r$ elements.

There are a number of shortcomings and mistakes in this preprint, that will be fixed in a new joint paper