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 $B^r_n$ 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 $CH^{(r-1)(n-1)} 查看全文>>