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We apply the filtered and graded methods developed in earlier works to find (noncommutative) free group algebras in division rings.
is a Lie algebra, we denote by
its universal enveloping algebra. P. M. Cohn constructed a division ring
. We denote by
the division subring of
be a field of characteristic zero, and let
be a nonabelian Lie
-algebra. If either
is residually nilpotent or
is an Ore domain, we show that
contains (noncommutative) free group algebras. In those same cases, if
is equipped with an involution, we are able to prove that the free group algebra in
can be chosen generated by symmetric elements in most cases.
be a nonabelian residually torsion-free nilpotent group, and let
be the division subring of the Malcev–Neumann series ring generated by the group algebra
is equipped with an involution, we show that
contains a (noncommutative) free group algebra generated by symmetric elements.
We show that over any field
of characteristic 2 and 2-rank
, there exist
-fold Pfister forms that have no slot in common. This answers a question of Becher [‘Triple linkage’, Ann.
-Theory, to appear] in the negative. We provide an analogous result also for quadratic Pfister forms.
In characteristic two, some criteria are obtained for a symmetric square-central element of a totally decomposable algebra with orthogonal involution, to be contained in an invariant quaternion subalgebra.
We develop a theory of commensurability of groups, of rings, and of modules. It allows us, in certain cases, to compare sizes of automorphism groups of modules, even when those are infinite. This work is motivated by the Cohen–Lenstra heuristics on class groups.
In this article the
-essential dimension of generic symbols over fields of characteristic
is studied. In particular, the
-essential dimension of the length
-symbol of degree
is bounded below by
when the base field is algebraically closed of characteristic
. The proof uses new techniques for working with residues in Milne–Kato
-cohomology and builds on work of Babic and Chernousov in the Witt group in characteristic 2. Two corollaries on
-symbol algebras (i.e, degree 2 symbols) result from this work. The generic
-symbol algebra of length
is shown to have
-essential dimension equal to
-torsion Brauer class. The second is a lower bound of
-essential dimension of the functor
. Roughly speaking this says that you will need at least
independent parameters to be able to specify any given algebra of degree
over a field of characteristic
and improves on the previously established lower bound of 3.
We use twisted sheaves and their moduli spaces to study the Brauer group of a scheme. In particular, we (1) show how twisted methods can be efficiently used to re-prove the basic facts about the Brauer group and cohomological Brauer group (including Gabber’s theorem that they coincide for a separated union of two affine schemes), (2) give a new proof of de Jong’s period-index theorem for surfaces over algebraically closed fields, and (3) prove an analogous result for surfaces over finite fields. We also include a reduction of all period-index problems for Brauer groups of function fields over algebraically closed fields to characteristic zero, which (among other things) extends de Jong’s result to include classes of period divisible by the characteristic of the base field. Finally, we use the theory developed here to give counterexamples to a standard type of local-to-global conjecture for geometrically rational varieties over the function field of the projective plane.
Abstract. For F a field we compute, explicitly and directly, the right Krull dimension of the algebra Qop⊗FQ for certain semisimple Artinian F-algebras Q. (Here Qop denotes the opposite ring of Q.) We use our calculation to give alternative proofs of some theorems of J. T. Stafford and A. I. Lichtman. Our methods involve a detailed study of skew polynomial rings.
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