Highly versal torsors

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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 languageAmerican English
Title of host publicationAmitsur Centennial Symposium, 2021
EditorsAvinoam Mann, Louis H. Rowen, David J. Saltman, Aner Shalev, Lance W. Small, Uzi Vishne
PublisherAmerican Mathematical Society
Pages129-174
Number of pages46
ISBN (Print)9781470475550
DOIs
StatePublished - 2024
EventAmitsur Centennial Symposium, 2021 - Jerusalem, Israel
Duration: 1 Nov 20214 Nov 2021

Publication series

NameContemporary Mathematics
Volume800

Conference

ConferenceAmitsur Centennial Symposium, 2021
Country/TerritoryIsrael
CityJerusalem
Period1/11/214/11/21

Keywords

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

All Science Journal Classification (ASJC) codes

  • General Mathematics

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