## 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.

Original language | English |
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Pages (from-to) | 7277-7321 |

Number of pages | 45 |

Journal | Transactions of the American Mathematical Society |

Volume | 375 |

Issue number | 10 |

DOIs | |

State | Published - 1 Oct 2022 |

### Bibliographical note

Funding Information:Received by the editors January 9, 2022, and, in revised form, March 7, 2022. 2020 Mathematics Subject Classification. Primary 14L30, 16H05, 16S15, 14F25, 55R40; Secondary 17A36, 13E15, 20G10, 14M17. Key words and phrases. Algebra over a ring, multialgebra, Azumaya algebra, number of generators, versal torsor, classifying space, equivariant cohomology. The second and third authors were partially supported by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada.

Publisher Copyright:

© 2022 American Mathematical Society.

## Keywords

- Algebra over a ring
- Azumaya algebra
- classifying space
- equivariant cohomology
- multialgebra
- number of generators
- versal torsor

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics