Translations of Mathematical Monographs最新文献

In this paper we discuss the Brauer group of a field and its connections with cohomology groups. Definitions involving central simple algebras lead to a discussion of splitting fields, which are the important step in the connection of the Brauer group with cohomology groups. Finally, once the connection between the Brauer group and cohomology groups is established, specific examples of cocycles associated to central simple algebras are calculated 1. Central Simple Algebras and Splitting Fields Elements of the Brauer group are equivalence classes of central simple algebras. As such, it is important to have an understanding of these algebras to understand the Brauer group. This section aims to lay the foundation for the rest of our discussion; this foundation starts with the definition of a CSA. Definitions 1.1: Fix some field k. 1. An algebra over k is a ring A along with an embedding ψ : k ↪→ A, where 1 in k maps to 1 in A. This embedding induces a scalar product that allows A to have a vector space structure over k. The image ψ(k) is often denoted k · 1, or simply k. We require ψ(k) to commute with every element of the algebra, so that the “left scalar product” and the “right scalar product” will be the same. 2. The center of an algebra A is the set Z(A) = {z ∈ A : az = za ∀a ∈ A}. If this set is the subspace k · 1, A is said to be central. 3. A (two-sided) ideal of an algebra is a (two-sided) ideal of the algebra viewed as a ring. Note that (x ·1)i = x · i ∈ I for every x ∈ k, so it has the addition stipulation of being closed under scalar multiplication. If the only ideals of A are {0} and A, A is said to be simple. 4. An algebra is a central simple algebra if it is both central and simple. Central simple algebras are often referred to as CSAs, an abbreviation we will use often. In addition to these definitions, it is useful to know when an algebra is finite-dimensional. An algebra is finite-dimensional when it is finite-dimensional as a vector space over k. All algebras will be assumed to be finite-dimensional unless stated otherwise. Examples 1.2: Common examples of CSAs include: 1. Mn(k) is a CSA over k for all n > 0. The matrix ring is equipped with a scalar product operation that multiplies each entry by an element of k. 2. Central division algebras, including k itself, are CSAs over k as well. 3. The quaternion algebra ( a,b k ) , generated by i and j with i2 = a, j2 = b, ij = −ji is a central simple algebra over k. In fact, it is either a division algebra or it is isomorphic to M2(k). A discussion of these algebras and their properties is found in [Lam].

本文讨论了域的Brauer群及其与上同调群的联系。引入中心简单代数的定义导致了对分裂场的讨论，这是Brauer群与上同调群连接的重要步骤。最后，一旦建立了Brauer群和上同群之间的联系，计算了与中心简单代数相关的环的具体例子1。中心简单代数与分裂域Brauer群的元是中心简单代数的等价类。因此，理解这些代数对于理解布劳尔群是很重要的。本节旨在为我们接下来的讨论奠定基础;这个基础从CSA的定义开始。定义1.1:修正一些字段k。一个代数k是一个环连同一个嵌入ψ:k↪→a, 1 k映射到1在这嵌入诱发一个标量产品允许有一个向量空间结构/ k。图像ψ(k)通常是表示k·1,或者只是k。我们需要ψ(k)通勤与代数的每一个元素,所以“左数积”和“正确的标量产品”将是相同的。2. 代数A的中心是集合Z(A) = {Z∈A: az = za∀A∈A}。如果这个集合是子空间k·1，我们说A是中心的。3.一个代数的(双面)理想是被看作一个环的代数的(双面)理想。注意，对于每一个x∈k， (x·1)i = x·i∈i，所以它具有标量乘法封闭的加法规定。如果A的唯一理想是{0}和A，则说A是简单的。4. 如果一个代数既是中心代数又是简单代数，那么它就是中心简单代数。中心简单代数通常被称为csa，这是我们经常使用的缩写。除了这些定义之外，知道代数何时是有限维的也是很有用的。当一个代数作为k上的向量空间是有限维时，它就是有限维的。除非另有说明，否则所有代数都被假定为有限维的。示例1.2:csa的常见示例包括:1;Mn(k)是所有n > 0的CSA除以k。矩阵环配备了一个标量乘积运算，它将每一项乘以k的一个元素。中心除法代数，包括k本身，也是csa除以k。3.由i和j生成i2 = a, j2 = b, ij = - ji的四元数代数(a,b k)是k上的一个中心简单代数。事实上，它要么是一个除法代数，要么是与M2(k)同构的。关于这些代数及其性质的讨论见[Lam]。

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Example of a Non-rational Stably Rational Variety 一个非理性的稳定理性变种的例子

Translations of Mathematical Monographs

Pub Date : 2018-09-10 DOI: 10.1090/mmono/246/09

引用次数: 0

Non-rational Double Covers of ℙ³ 非理性的双罩

Translations of Mathematical Monographs

Pub Date : 2018-09-10 DOI: 10.1090/mmono/246/07

引用次数: 0

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