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

First, Uriya; Reichstein, Zinovy; Williams, Ben

doi:10.1090/tran/8720

On the number of generators of an algebra over a commutative ring
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by Uriya A. First, Zinovy Reichstein and Ben Williams PDF

Trans. Amer. Math. Soc. 375 (2022), 7277-7321 Request permission

Abstract:

A theorem of O. Forster says that if $R$ is a noetherian ring of Krull dimension $d$, then every 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 rank-$n$ projective $R$-modules as $R$-forms of the non-unital $R$-algebra $R^n$ 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 finite étale algebras.

In this paper, we prove new upper and lower bounds on the number of generators of an $R$-form of a $k$-algebra, where $k$ is an infinite field and $R$ is a finitely generated $k$-ring of Krull dimension $d$. 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 type.

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