The equivariant Brauer group of a group

Compact Schur-Weyl duality: real Lie groups and the cyclotomic Brauer algebra

We show that the centraliser of the maximal compact subgroup of the real orthogonal or symplectic groups acting on tensors of their standard representation are isomorphic to cyclotomic Brauer algebras. We also show that for the symplectic group this cyclotomic Brauer algebra splits into summands of walled Brauer algebras.

Galois-Azumaya Extensions and the Brauer-Galois Group of a Commutative Ring

We attach to any commutative ring R a subgroup of the Brauer group of R, called the Brauer-Galois group of R. Its elements are the classes of the Azumaya R-algebras which can be represented, via Brauer equivalence, by a Galois extension of R. We compute this group for some particular commutative fields.

Severi-Brauer簇自同构群的有限子群

Finite subgroups of automorphism groups of Severi--Brauer varieties

本文研究极小Severi-Brauer簇上有限子群的结构，并完整描述了维数为素数减一的这类簇上的所有有限子群。

Brauer groups of genus zero extensions of number fields

We determine the isomorphism class of the Brauer groups of certain nonrational genus zero extensions of number fields. In particular, for all genus zero extensions E of the rational numbers Q that are split by Q(\sqrt{2}), Br(E) is isomorphic to Br(Q(t)).

The Brauer group of Burnside rings

The Brauer group of a commutative ring is an important invariant of a commutative ring, a common journeyman to the group of units and the Picard group. Burnside rings of finite groups play an important role in representation theory, and their groups of units and Picard groups have been studied extensively. In this short note, we completely determine the Brauer groups of Burnside rings: they vanish.

Hasse Invariant for the Tame Brauer Group of a Higher Local Field

We generalize the Hasse invariant of local class field theory to the tame Brauer group of a higher dimensional local field, and use it to study the arithmetic of central simple algebras over such fields, which are given {\it a priori} as tensor products of standard cyclic algebras. We also compute the tame Brauer dimension (or {\it period-index bound}) and the cyclic length of a general henselian-valued field of finite rank and finite residue field.

Corestriction for algebras with group action

We define a corestriction map for equivariant Brauer groups in the sense of Fröhlich and Wall, which contain as a special case the Brauer-Clifford groups introduced by Turull. We show that this corestriction map has similar properties as the corestriction map in group cohomology (especially Galois cohomology). In particular, composing corestriction and restriction associated to a subgroup $H\leq G$ amounts to powering with the index $\lvert G:H \rvert$.

Relative Brauer groups of genus 1 curves

In this paper we develop techniques for computing the relative Brauer group of curves, focusing particularly on the case where the genus is 1. We use these techniques to show that the relative Brauer group may be infinite (for certain ground fields) as well as to determine this group explicitly for certain curves defined over the rational numbers. To connect to previous descriptions of relative Brauer groups in the literature, we describe a family of genus 1 curves, which we call "cyclic type" for which the relative Brauer group can be shown to have a particularly nice description. In order to…

A Quillen model for classical Morita theory and a tensor categorification of the Brauer group

Let K be a commutative ring. In this article we construct a symmetric monoidal Quillen model structure on the category of small K-categories which enhances classical Morita theory. We then use it in order to obtain a natural tensor categorification of the Brauer group and of its functoriality.

Cohomology of Restricted Lie-Rinehart Algebras and the Brauer Group

We give an interpretation of the Brauer group of a purely inseparable extension of exponent 1, in terms of restricted Lie-Rinehart cohomology. In particular, we define and study the category $p$-$\rm{LR}(A)$ of restricted Lie-Rinehart algebras over a commutative algebra $A$. We define cotriple cohomology groups $H_{p-LR}(L,M)$ for $L\in p$-$\rm{LR}(A)$ and $M$ a Beck $L$-module. We classify restricted Lie-Rinehart extensions. Thus, we obtain a classification theorem for regular extensions considered by Hoshschild.

Brauer groups and étale cohomology in derived algebraic geometry

In this paper, we study Azumaya algebras and Brauer groups in derived algebraic geometry. We establish various fundamental facts about Brauer groups in this setting, and we provide a computational tool, which we use to compute the Brauer group in several examples. In particular, we show that the Brauer group of the sphere spectrum vanishes, and we use this to prove two uniqueness theorems for the stable homotopy category. Our key technical results include the local geometricity, in the sense of Artin n-stacks, of the moduli space of perfect modules over a smooth and proper algebra, the étale l…

Brauer groups for commutative $S$-algebras

We investigate a notion of Azumaya algebras in the context of structured ring spectra and give a definition of Brauer groups. We investigate their Galois theoretic properties, and discuss examples of Azumaya algebras arising from Galois descent and cyclic algebras. We construct examples that are related to topological Hochschild cohomology of group ring spectra and we present a K(n)-local variant of the notion of Brauer groups.

The genus of a division algebra and the unramified Brauer group

Let D be a finite-dimensional central division algebra over a field K. We define the genus gen(D) of D to be the collection of classes in the Brauer group of K represented by central division K-algebras D' having the same maximal subfields as D. In this paper, we describe a general approach to proving the finiteness of gen(D) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of K.

A sequence to compute the Brauer group of certain quasi-triangular Hopf algebras

A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra $B$ in a braided monoidal category $\C$, and under certain assumptions on the braiding (fulfilled if $\C$ is symmetric), we construct a sequence for the Brauer group $\BM(\C;B)$ of $B$-module algebras, generalizing Beattie's one. It allows one to prove that $\BM(\C;B) \cong \Br(\C) \times \Gal(\C;B),$ where $\Br(\C)$ is the Brauer group of $\C$ and $\Gal(…