Highly Versal Torsors

TL;DR

The paper introduces highly versal torsors with stronger specialization properties over various schemes, and applies these results to establish uniform bounds on the symbol length of certain Azumaya algebras.

Contribution

It constructs highly versal torsors with enhanced specialization capabilities and demonstrates their applications to bounding the symbol length of Azumaya algebras.

Findings

Existence of torsors that specialize to all torsors over quasi-projective schemes after removing codimension-d subsets.

Every globally generated vector bundle over a finite type scheme can be generated by n+d global sections.

Bounded symbol length for degree-m, period-n Azumaya algebras over certain rings.

Abstract

Let $G$ be a linear algebraic group over an infinite field $k$. Loosely speaking, a $G$-torsor over $k$-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 $d\in\mathbb{N}$, there exists a $G$-torsor over a smooth quasi-projective $k$-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 $\operatorname{char} k=0$. We apply our highly versal torsors to show that, for fixed $m,n\in\mathbb{N}$, the symbol length of any degree-$m$ period-$n$ Azumaya algebra over any local $\mathbb{Z}[\frac{1}{n},e^{2\pi i/n}]$-ring is uniformly bounded. A similar statement holds in the semilocal case, but under mild restrictions on the base ring.