Bounding the minimal number of generators of an Azumaya algebra

TL;DR

This paper investigates the minimal number of generators needed for Azumaya algebras over rings of a given dimension, providing explicit examples and showing that no uniform bound exists for all cases.

Contribution

It constructs specific Azumaya algebras requiring a calculated number of generators, demonstrating the non-existence of a universal upper bound on generators needed.

Findings

Explicit examples of Azumaya algebras with minimal generators

Demonstration that the number of generators can grow with ring dimension

Identification of characteristic class obstructions to generation

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 $B^r_n$ 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 $CH^{(r-1)(n-1)}_{PGL_n} \to CH^{(r-1)(n-1)}(B^r_n)$ 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.