26 research outputs found
Highly Versal Torsors
Let be a linear algebraic group over an infinite field . Loosely
speaking, a -torsor over -variety is said to be versal if it specializes
to every -torsor over any -field. The existence of versal torsors is
well-known. We show that there exist -torsors that admit even stronger
versality properties. For example, for every , there exists a
-torsor over a smooth quasi-projective -scheme that specializes to every
torsor over a quasi-projective -scheme after removing some codimension-
closed subset from the latter. Moreover, such specializations are abundant in a
well-defined sense. Similar results hold if we replace with an arbitrary
base-scheme. In the course of the proof we show that every globally generated
rank- vector bundle over a -dimensional -scheme of finite type can be
generated by global sections.
When can be embedded in a group scheme of unipotent upper-triangular
matrices, we further show that there exist -torsors specializing to every
-torsor over any affine -scheme. We show that the converse holds when
.
We apply our highly versal torsors to show that, for fixed
and a fixed semilocal -ring with
infinite residue fields, the symbol length of Azumaya algebras over having
degree and period is uniformly bounded. Under mild assumptions on ,
e.g., if is local, the bound depends only on and , and not on .Comment: 41 pages. Comments are welcom
Rationally Isomorphic Hermitian Forms and Torsors of Some Non-Reductive Groups
Let be a semilocal Dedekind domain. Under certain assumptions, we show
that two (not necessarily unimodular) hermitian forms over an -algebra with
involution, which are rationally ismorphic and have isomorphic semisimple
coradicals, are in fact isomorphic. The same result is also obtained for
quadratic forms equipped with an action of a finite group. The results have
cohomological restatements that resemble the Grothendieck--Serre conjecture,
except the group schemes involved are not reductive. We show that these group
schemes are closely related to group schemes arising in Bruhat--Tits theory.Comment: 27 pages. Changes from previous version: Section 5 was split into two
sections, several proofs have been simplified, other mild modification