Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On the number of generators of an algebra over a commutative ring
HTML articles powered by AMS MathViewer

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.

References
Similar Articles
Additional Information
  • Uriya A. First
  • Affiliation: Department of Mathematics, Faculty of Natural Sciences, University of Haifa, Mount Carmel, Haifa 3498838, Israel
  • MR Author ID: 1007314
  • ORCID: 0000-0002-6754-8009
  • Email: ufirst@univ.haifa.ac.il
  • Zinovy Reichstein
  • Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
  • MR Author ID: 268803
  • ORCID: 0000-0002-3157-2066
  • Email: reichst@math.ubc.ca
  • Ben Williams
  • Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
  • MR Author ID: 924943
  • ORCID: 0000-0001-9688-2471
  • Email: tbjw@math.ubc.ca
  • Received by editor(s): January 9, 2022
  • Received by editor(s) in revised form: March 7, 2022
  • Published electronically: August 11, 2022
  • Additional Notes: The second and third authors were partially supported by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 7277-7321
  • MSC (2020): Primary 14L30, 16H05, 16S15, 14F25, 55R40; Secondary 17A36, 13E15, 20G10, 14M17
  • DOI: https://doi.org/10.1090/tran/8720