Brauer groups of genus zero extensions of number fields

On the Brauer groups of quasilocal fields and the norm groups of their finite Galois extensions

This paper shows that divisible abelian torsion groups are realizable as Brauer groups of quasilocal fields. It describes the isomorphism classes of Brauer groups of primarily quasilocal fields and solves the analogous problem concerning the reduced components of the Brauer groups of two basic types of Henselian valued absolutely stable fields. For a quasilocal field E and a finite separable extension R/E, we find two sufficient conditions for validity of the norm group equality $N(R/E) = N(R_{0}/E)$, where R_{0} is the maximal abelian extension of E in R. This is used for deriving information…

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.

单项代数的平凡扩展是称性分数布劳尔配置代数

Trivial extensions of monomial algebras are symmetric fractional Brauer configuration algebras

本文通过等价定义建立了单项代数与分数布劳尔配置代数的关系，证明了其与平凡扩展的同构性，并建立了单项代数与称性分数布劳尔配置代数的对应关系。

On Brauer $p$-dimensions and index-exponent relations over finitely-generated field extensions

Let $E$ be a field of absolute Brauer dimension abrd$(E)$, and $F/E$ a transcendental finitely-generated extension. This paper shows that the Brauer dimension Brd$(F)$ is infinite, if abrd$(E) = \infty $. When the absolute Brauer $p$-dimension abrd$_{p}(E)$ is infinite, for some prime number $p$, it proves that for each pair $(n, m)$ of integers with $n \ge m > 0$, there is a central division $F$-algebra of Schur index $p ^{n}$ and exponent $p ^{m}$. Lower bounds on the Brauer $p$-dimension Brd$_{p}(F)$ are obtained in some important special cases where abrd$_{p}(E) < \infty $. These results s…

通过覆盖提升 $A_\infty$-结构及其在分数 Brauer 图代数中的应用

Lifting $A_\infty$-structures through coverings with applications to fractional Brauer graph algebras

本文研究 $A_\infty$-结构在覆盖下的提升问题，证明相容群分次决定唯一典范提升，并应用于构造分数 Brauer 图代数的 $A_\infty$-范畴，建立导出等价与组合不变量。

椭圆曲线及其主齐性空间：分裂Severi--Brauer簇

Elliptic curves and their principal homogeneous spaces: splitting Severi--Brauer varieties

本文研究哪些椭圆曲线可作为分裂Severi--Brauer簇的亏格一曲线的雅可比簇，给出三个新例子：刻画了完美域扩张下主齐性空间分裂Severi--Brauer簇的循环代数条件，构造了次数最小且不可再降的嵌入例子，并将Saltman关于曲面的结果推广到任意偶数维。

On the size of the genus of a division algebra

Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) of D as the set of classes [D'] in the Brauer group Br(K) of K represented by central division algebras D' of degree n over K having the same maximal subfields as D. We prove that if the field K is finitely generated and n is prime to its characteristic, then gen(D) is finite, and give explicit estimations of its size in certain situations.

On the behaviour of Brauer $p$-dimensions under finitely-generated field extensions

The present paper shows that if $q \in \mathbb P$ or $q = 0$, where $\mathbb P$ is the set of prime numbers, then there exist characteristic $q$ fields $E _{q,k}\colon \ k \in \mathbb N$, of Brauer dimension Brd$(E _{q,k}) = k$ and infinite absolute Brauer $p$-dimensions abrd$_{p}(E _{q,k})$, for all $p \in \mathbb P$ not dividing $q ^{2} - q$. This ensures that Brd$_{p}(F _{q,k}) = \infty $, $p \dagger q ^{2} - q$, for every finitely-generated transcendental extension $F _{q,k}/E _{q,k}$. We also prove that each sequence $a _{p}, b _{p}$, $p \in \mathbb P$, satisfying the conditions $a _{2} =…

Abelian extensions of global fields with constant local degrees

Given a global field K and a positive integer n, there exists an abelian extension L/K (of exponent n) such that the local degree of L/K is equal to n at every finite prime of K, and is equal to two at the real primes if n=2. As a consequence, the n-torsion subgroup of the Brauer group of K is equal to the relative Brauer group of L/K.

Division algebras of prime degree with infinite genus

The genus gen(D) of a finite-dimensional central division algebra D over a field F is defined as the collection of classes [D'] in the Brauer group Br(F), where D' is a central division F-algebra having the same maximal subfields as D. For any prime p, we construct a division algebra of degree p with infinite genus. Moreover, we show that there exists a field K such that there are infinitely many nonisomorphic central division K-algebras of degree p, and any two such algebras have the same genus.

Witt kernels and Brauer kernels for quartic extensions in characteristic two

Let $F$ be a field of characteristic $2$ and let $E/F$ be a field extension of degree $4$. We determine the kernel $W_q(E/F)$ of the restriction map $W_qF\to W_qE$ between the Witt groups of nondegenerate quadratic forms over $F$ and over $E$, completing earlier partial results by Ahmad, Baeza, Mammone and Moresi. We also deduct the corresponding result for the Witt kernel $W(E/F)$ of the restriction map $WF\to WE$ between the Witt rings of nondegenerate symmetric bilinear forms over $F$ and over $E$ from earlier results by the first author. As application, we describe the $2$-torsion part of …

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.

Lower bounds and infinity criterion for Brauer $p$-dimensions of finitely-generated field extensions

Let $E$ be a field, $p$ a prime number and $F/E$ a finitely-generated extension of transcendency degree $t$. This paper shows that if the absolute Galois group $\mathcal{G}_{E}$ is of nonzero cohomological $p$-dimension cd$_{p}(E)$, then the field $F$ has Brauer $p$-dimension Brd$_{p}(F) \ge t$ except, possibly, in case $p = 2$, the Sylow pro-2-subgroups of $\mathcal{G}_{E}$ are of order 2, and $F$ is a nonreal field. It announces that Brd$_{p}(F)$ is infinite whenever $t \ge 1$ and the absolute Brauer $p$-dimension abrd$_{p}(E)$ is infinite; moreover, for each pair $(m, n)$ of integers with $…

Index reduction for Brauer classes via stable sheaves (with an appendix by Bhargav Bhatt)

We use twisted sheaves to study the problem of index reduction for Brauer classes. In general terms, this problem may be phrased as follows: given a field $k$, a $k$-variety $X$, and a class $α\in \Br(k)$, compute the index of the class $α_{k(X)} \in \Br(X)$ obtained from $α$ by extension of scalars to $k(X)$. We give a general method for computing index reduction which refines classical results of Schofield and van den Bergh. When $X$ is a curve of genus 1, we use Atiyah's theorem on the structure of stable vector bundles with integral slope to show that our formula simplifies dramatically, g…

斜Brauer图代数

Skew-Brauer graph algebras

本文提出斜Brauer图代数，推广Brauer图代数，并研究其表示类型与几何解释。

Chow groups of products of Severi-Brauer varieties and invariants of degree 3

We study the semi-decomposable invariants of a split semisimple group and their extension to a split reductive group by using the torsion in the codimension $2$ Chow groups of a product of Severi-Brauer varieties. In particular, for any $n\geq 2$ we completely determine the degree $3$ invariants of a split semisimple group, the quotient of $(\operatorname{\mathbf{SL}}_{2})^{n}$ by its maximal central subgroup, as well as of the corresponding split reductive group. We also provide an example illustrating that a modification of our method can be applied to find the semi-decomposable invariants o…

无限超越度域上除代数的亏格

Genus of division algebras over fields with infinite transcendence degree

本文证明了在无限超越度纯超越扩张的有限扩张域上，有限维中心除代数的亏格有限，并进一步给出指数为2的除代数及G2型单代数群的亏格均为平凡的结论。

Compact Schur-Weyl duality and the affine Type B/C Brauer algebra

We define an extension of the affine Brauer algebra, the type B/C affine Brauer algebra. This new algebra contains the hyperoctahedral group and it naturally acts on $END_K(X \otimes V^{\otimes k})$ for Orthogonal and Symplectic groups. Thus we obtain a compact analogue of Schur-Weyl duality. We study functors $F_{μ,k}$ from the category of admissible $O(p,q)$ or $Sp_{2n}(\mathbb{R})$ modules to representations of the type B/C affine Brauer algebra $\mathfrak{B}_k^θ$. Thus providing a Akawaka-Suzuki-esque link between $O(p,q)$ (or $Sp_{2n}(\mathbb{R})$) and $\mathfrak{B}_k^θ$. Furthermore thes…

Primarily quasilocal fields and 1-dimensional abstract local class field theory

Let $E$ be a field satisfying the following conditions: (i) the $p$-component of the Brauer group Br$(E)$ is nontrivial whenever $p$ is a prime number for which $E$ is properly included in its maximal $p$-extension; (ii) the relative Brauer group Br$(L/E)$ equals the maximal subgroup of Br$(E)$ of exponent $p$, for every cyclic extension $L/E$ of degree $p$. The paper proves that finite abelian extensions of $E$ are uniquely determined by their norm groups and related essentially as in the classical local class field theory. This includes analogues to the fundamental correspondence, the local …

Orthogonal involutions on central simple algebras and function fields of Severi-Brauer varieties

An orthogonal involution $σ$ on a central simple algebra $A$, after scalar extension to the function field $\mathcal{F}(A)$ of the Severi--Brauer variety of $A$, is adjoint to a quadratic form $q_σ$ over $\mathcal{F}(A)$, 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 $q_σ$. As opposed to this, we prove that there exists non-totally decomposable orthogonal involuti…

Mixed Characteristic Cyclic Matters

The Artin-Schreier polynomial $Z^p - Z - a$ is very well known. Polynomials of this type describe all degree $p$ (cyclic) Galois extensions over any commutative ring of characteristic $p$. Equally attractive is the associated Galois action. If $θ$ is a root then $σ(θ) = θ+ 1$ generates the Galois group. Less well known, but equally general, is the so called "differential crossed product" Azumaya algebra generated by $x,y$ subject to the relation $xy - yx = 1$. In characteristic $p$ these algebras are always Azumaya and algebras of this sort generate the $p$ torsion subgroup of the Brauer group…

Cocycle twisting of E(n)-module algebras and applications to the Brauer group

We classify the orbits of coquasi-triangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group. This is applied to detect subgroups of the Brauer group $BQ(k,E(n))$ of E(n) that are isomorphic. For a triangular structure $R$ on E(n) we prove that the subgroup $BM(k,E(n),R)$ of $BQ(k,E(n))$ arising from $R$ is isomorphic to a direct product of $BW(k)$, the Brauer-Wall group of the ground field $k$, and $Sym_n(k)$, the group of $n \times n$ symmetric matrices under addition. For a general quasi-triangular structure $R'$ on E(n) we construct a …

关于不完全剩余域情形的Milnor $K$-理论及其在周期-指数问题中的应用

On Milnor $K$-theory in the imperfect residue case and applications to period-index problems

本文通过定义伪完美扩张，证明其可分裂混合特征完备离散赋值域的高阶上同调类，从而给出Brauer p-维数上界，并改进半全局域上的周期-指数界。

Higher genus affine algebras of Krichever - Novikov type

For higher genus multi-point current algebras of Krichever-Novikov type associated to a finite-dimensional Lie algebra, local Lie algebra two-cocycles are studied. They yield as central extensions almost-graded higher genus affine Lie algebras. In case that the Lie algebra is reductive a complete classification is given. For a simple Lie algebra, like in the classical situation, there is up to equivalence and rescaling only one non-trivial almost-graded central extension. The classification is extended to the algebras of meromorphic differential operators of order less or equal one on the curr…