@inproceedings{843368077ae5456092fc228df190b7f6,

title = "Highly versal torsors",

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.",

keywords = "Azumaya algebra, Galois extension, Group scheme, linear algebraic group, principal homogeneous space, symbol length, torsor, vector bundle, versal torsor",

author = "First, {Uriya A.}",

note = "Publisher Copyright: {\textcopyright} 2024 Uriya A. First.; Amitsur Centennial Symposium, 2021 ; Conference date: 01-11-2021 Through 04-11-2021",

year = "2024",

doi = "https://doi.org/10.1090/conm/800/16054",

language = "American English",

isbn = "9781470475550",

series = "Contemporary Mathematics",

publisher = "American Mathematical Society",

pages = "129--174",

editor = "Avinoam Mann and Rowen, {Louis H.} and Saltman, {David J.} and Aner Shalev and Small, {Lance W.} and Uzi Vishne",

booktitle = "Amitsur Centennial Symposium, 2021",

address = "United States",

}