On the number of generators of an algebra over a commutative ring

TL;DR

This paper investigates the minimal number of generators needed for algebraic structures over rings, extending classical bounds and revealing that these bounds are often not tight, especially for Azumaya algebras, using geometric classifying space methods.

Contribution

It establishes new bounds on generators for algebra forms over rings with finite transcendence degree, showing classical bounds are often not optimal for many algebra types.

Findings

Classical Forster bounds are not tight for most algebra forms.

New bounds are derived for algebras over rings with finite transcendence degree.

Results are especially detailed for Azumaya algebras.

Abstract

A theorem of O. Forster says that if $R$ is a noetherian ring of Krull dimension $d$, then any projective $R$-module of rank $n$ can be generated by $d+n$ elements. S. Chase and R. Swan subsequently showed that this bound is sharp: there exist examples that cannot be generated by fewer than $d+n$ elements. We view projective $R$-modules as $R$-forms of the non-unital $R$-algebra where the product of any two elements is $0$. The first two authors generalized Forster's theorem to forms of other algebras (not necessarily commutative, associative or unital); A. Shukla and the third author then showed that this generalized Forster bound is optimal for \'etale algebras. In this paper, we prove new upper and lower bound on the number of generators of an $R$-form of a $k$-algebra, where $k$ is an infinite field and $R$ has finite transcendence degree $d$ over $k$. In particular, we show that, contrary to expectations, for most types of algebras, the generalized Forster bound is far from optimal. Our results are particularly detailed in the case of Azumaya algebras. Our proofs are based on reinterpreting the problem as a question about approximating the classifying stack $BG$, where $G$ is the automorphism group of the algebra in question, by algebraic spaces of a certain form.