We introduce a cohomology theory that classifies differential objects that arise from Picard-Vessiot theory, using differential Hopf-Galois descent. To do this, we provide an explicit description of ...Picard-Vessiot theory in terms of differential torsors. We then use this cohomology to give a bijective correspondence between differential objects and differential torsors. As an application, we prove a universal bound for the differential splitting degree of differential central simple algebras.
An associative central simple algebra is a form of a matrix algebra, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and ...isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrix algebras, which we are led to define and investigate. Semiassociative algebras modulo skew matrix algebras compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup.
It is well known that central simple algebras are split by suitable finite Galois extensions of their centers. In 4 a counterpart of this result was studied in the set up of differential matrix ...algebras, wherein Picard-Vessiot extensions that split matrix differential algebras were constructed. In this article, we exhibit instances of differential matrix algebras which are split by finite extensions. In some cases, we relate the existence of finite splitting extensions of a differential matrix algebra to the triviality of its tensor powers, and show in these cases, that orders of differential matrix algebras divide their degrees.
We show that any adjoint absolutely simple linear algebraic group over a field of characteristic zero is the automorphism group of some projector on a central simple algebra. Projective homogeneous ...varieties can be described in these terms; in particular, we reproduce quadratic equations by Lichtenstein defining them.
Let G be a split simple affine algebraic group of type A or C over a field k, and let E be a standard generic G-torsor over a field extension of k. We compute the Chow ring of the variety of Borel ...subgroups of G (also called the variety of complete flags of G), twisted by E. In most cases, the answer contains a large finite torsion subgroup. The torsion-free cases have been treated in the predecessor Chow ring of some generically twisted flag varieties by the author.
We extend, to the case of local fields of even characteristic, our previous computations for the set of maximal orders containing given quaternions as concrete sub-graphs of the Bruhat-Tits tree. ...Computing the relative position of such sub-graphs, the branches, is useful, for instance, to explicitly compute relative spinor images, thus solving the selectivity problem. They also play a role in the description of quotient graphs, which are useful to study matrix groups of arithmetical importance. As in earlier work, we concentrate on orders generated by two quaternions, but extending our method to larger generating sets is quite straightforward. In our previous work, the results were given mainly in terms of the quadratic defect. In the present context, we introduce and characterize an analogous concept for Artin-Scheier extensions, to take care of quaternions generating Galois field extensions. The latter extensions have no non-trivial elements of null trace. For this reason, we state our results in terms of a wider family of algebra presentations, in contrast to the aforementioned work, where we chose a pair of pure quaternions as generators of the order.
Branches on division algebras Arenas, Manuel; Arenas-Carmona, Luis
Journal of number theory,
January 2020, 2020-01-00, Volume:
206
Journal Article
Peer reviewed
Open access
We describe the set of maximal orders in a 2-by-2 matrix algebra over a non-commutative local division algebra B containing a given suborder, for certain important families of such suborders, ...including rings of integers of division subalgebras of B or most maximal semisimple commutative subalgebras.
An orthogonal involution σ on a central simple algebra A, after scalar extension to the function field F(A) of the Severi–Brauer variety of A, is adjoint to a quadratic form qσ over F(A), which is ...uniquely defined up to a scalar factor. Some properties of the involution, such as hyperbolicity, and isotropy up to an odd-degree extension of the base field, are encoded in this quadratic form, meaning that they hold for the involution σ if and only if they hold for qσ. As opposed to this, we prove that there exists non-totally decomposable orthogonal involutions that become totally decomposable over F(A), so that the associated form qσ is a Pfister form. We also provide examples of nonisomorphic involutions on an index 2 algebra that yield similar quadratic forms, thus proving that the form qσ does not determine the isomorphism class of σ, even when the underlying algebra has index 2. As a consequence, we show that the e3 invariant for orthogonal involutions is not classifying in degree 12, and does not detect totally decomposable involutions in degree 16, as opposed to what happens for quadratic forms.
In this manuscript, it is shown that the group of K1-zero-cycles on the second generalized Severi–Brauer variety of an algebra A of index 4 is given by elements of the group K1(A) together with a ...square-root of their reduced norm. Utilizing results of Krashen concerning exceptional isomorphisms, we translate our problem to the computation of cycles on involution varieties. Work of Chernousov and Merkurjev then gives a means of describing such cycles in terms of Clifford and spin groups and corresponding R-equivalence classes. We complete our computation by giving an explicit description of these algebraic groups.
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