## Abstract

Let G be a linear algebraic group over an infinite field k. Loosely speaking, a G-torsor over a k-variety is said to be versal if it specializes to every G-torsor over any k-field. The existence of versal torsors is well-known. We show that there exist G-torsors that admit even stronger versality properties. For example, for every d ∈ N, there exists a G-torsor over a smooth quasiprojective k-scheme that specializes to every torsor over a quasi-projective k-scheme after removing some codimension-d closed subset from the latter. Moreover, such specializations are abundant in a well-defined sense. Similar results hold if we replace k with an arbitrary base-scheme. In the course of the proof we show that every globally generated rank-n vector bundle over a d-dimensional k-scheme of finite type can be generated by n+d global sections. When G can be embedded in a group scheme of unipotent upper-triangular matrices, we further show that there exist G-torsors specializing to every G-torsor over any affine k-scheme. We show that the converse holds when char k=0. We apply our highly versal torsors to show that, for fixed m, n ∈ N, the symbol length of any degree-m period-n Azumaya algebra over any local Z[1 n, e2πi/n]-ring is uniformly bounded. A similar statement holds in the semilocal case, but under mild restrictions on the base ring.

Original language | English |
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Title of host publication | Amitsur Centennial Symposium, 2021 |

Editors | Avinoam Mann, Louis H. Rowen, David J. Saltman, Aner Shalev, Lance W. Small, Uzi Vishne |

Publisher | American Mathematical Society |

Pages | 129-174 |

Number of pages | 46 |

ISBN (Print) | 9781470475550 |

DOIs | |

State | Published - 2024 |

Event | Amitsur Centennial Symposium, 2021 - Jerusalem, Israel Duration: 1 Nov 2021 → 4 Nov 2021 |

### Publication series

Name | Contemporary Mathematics |
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Volume | 800 |

ISSN (Print) | 0271-4132 |

ISSN (Electronic) | 1098-3627 |

### Conference

Conference | Amitsur Centennial Symposium, 2021 |
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Country/Territory | Israel |

City | Jerusalem |

Period | 1/11/21 → 4/11/21 |

### Bibliographical note

Publisher Copyright:© 2024 Uriya A. First.

## Keywords

- Azumaya algebra
- Galois extension
- Group scheme
- linear algebraic group
- principal homogeneous space
- symbol length
- torsor
- vector bundle
- versal torsor

## ASJC Scopus subject areas

- General Mathematics