Highly versal torsors

Author:

First Uriya

Abstract

Let G G be a linear algebraic group over an infinite field k k . Loosely speaking, a G G -torsor over a k k -variety is said to be versal if it specializes to every G G -torsor over any k k -field. The existence of versal torsors is well-known. We show that there exist G G -torsors that admit even stronger versality properties. For example, for every d N d\in \mathbb {N} , there exists a G G -torsor over a smooth quasi-projective k k -scheme that specializes to every torsor over a quasi-projective k k -scheme after removing some codimension- d d closed subset from the latter. Moreover, such specializations are abundant in a well-defined sense. Similar results hold if we replace k k with an arbitrary base-scheme. In the course of the proof we show that every globally generated rank- n n vector bundle over a d d -dimensional k k -scheme of finite type can be generated by n + d n+d global sections.

When G G can be embedded in a group scheme of unipotent upper-triangular matrices, we further show that there exist G G -torsors specializing to every G G -torsor over any affine k k -scheme. We show that the converse holds when c h a r k = 0 chark=0 .

We apply our highly versal torsors to show that, for fixed m , n N m,n\in \mathbb {N} , the symbol length of any degree- m m period- n n Azumaya algebra over any local Z [ 1 n , e 2 π i / n ] \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.

Publisher

American Mathematical Society

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