ESSENTIAL DIMENSION
ESSENTIAL DIMENSION
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基本维度
DOI：
10.1090/conm/493/09676
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
A. Merkurjev
中科院分区：
文献类型：
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作者：
A. Merkurjev
代数对象的本质维度是衡量对象复杂性的整数。例如，设Q=(a，b)K是特征不为2的基域F的域扩张K上的四元数代数，即Q是K上的基为{1，i，j，ij}的四维代数，乘法表i2=a，j2=b，ij=−ji，其中a和b是K中的非零元素。因此，Q由两个参数a和b决定。等价地，我们可以说代数Q定义在K的子域K‘=F(a，b)上，即，四元数代数Q‘=(a，b)K’在至多2次超越度的域K‘上Q≃Q’⊗K‘K.另一方面，如果a和b是F的代数闭包Falg上的两个变量，且K=Falg(a，b)，则Q=(a，b)K是K上的一个除法代数，它不能定义在K上至多有1个超越度的子域K‘上，因为根据Tsen定理，在子域K Falg⊂K上没有除法四元数代数。我们说这类四元数代数类的本质维度等于2。从非正式的角度讲，一类代数对象的本质维度是定义类中任何对象所需的代数无关参数的最小数目。本质维的概念是由J.Buhler和Z.Reichstein在[7]中对给定Galois群G的有限Galois域扩张类提出的，后来又在[37]中将其推广到线性代数群G的主齐性G-空间类。许多经典的代数对象，如单代数、二次型和Hermitian型、对合代数、Cayley代数、Jordan代数等，都与代数群的主齐性空间密切相关。例如，给出一个四元数代数与给出群PGL2的主要齐次空间是相同的。定义本质维度所需的一类代数对象的唯一性质是，对于每个域扩张K/F，我们有一个对象(例如四元数代数)的同构类的集合F(K)，并且对于F上的每个域同态K→L，场映射F(K)→F(L)的变化(例如，四元数代数的场运算Q 7→Q⊗K L的变化)。换句话说，F是从F的域扩张的范畴FieldsF到集合的范畴集的函子。文[5]定义了任意函子域F-→集的本质维度。基本维度的应用之一如下。假设我们想要检查由函子F给出的关于对象类的分类猜想是否成立。通常，分类猜想假定另一个函子L(a
The essential dimension of an algebraic object is an integer that measures the complexity of the object. For example, let Q = (a, b)K be a quaternion algebra over a field extension K of a base field F of characteristic not 2. That is, let Q be a 4-dimensional algebra over K with basis {1, i, j, ij} and multiplication table i2 = a, j2 = b, ij = −ji, where a and b are nonzero elements in K. Thus, Q is determined by two parameters a and b. Equivalently, we can say that the algebra Q is defined over the subfield K ′ = F (a, b) of K, namely, Q ≃ Q′ ⊗K′ K for the quaternion algebra Q′ = (a, b)K′ over the field K ′ of transcendence degree at most 2 over F . On the other hand, if a and b are two variables over an algebraic closure Falg of F and K = Falg(a, b), then Q = (a, b)K is a division algebra over K that cannot be defined over a subfield K ′ of K of transcendence degree at most 1 over F since by Tsen’s theorem, there are no division quaternion algebras over the subfield K Falg ⊂ K. We say that the essential dimension of the class of quaternion algebras is equal to 2. Informally speaking, the essential dimension of a class of algebraic objects is the minimal number of algebraically independent parameters one needs to define any object in the class. The notion of the essential dimension was introduced by J. Buhler and Z. Reichstein in [7] for the class of finite Galois field extensions with a given Galois group G and later in [37], it was extended to the class of principal homogeneous G-spaces for a linear algebraic group G. Many classical algebraic objects, such as simple algebras, quadratic and hermitian forms, algebras with involutions, Cayley algebras, Jordan algebras, etc., are closely related to the principal homogeneous spaces of algebraic groups. For example, to give a quaternion algebra is the same as to give a principal homogeneous space of the group PGL2. The only property of a class of algebraic objects needed to define the essential dimension is that for every field extension K/F , we have a set F(K) of isomorphism classes of objects (quaternion algebras, for example), and for every field homomorphism K → L over F , a change of field map F(K)→ F(L) (e.g., the change of field operation Q 7→ Q ⊗K L for quaternion algebras). In other words, F is a functor from the category FieldsF of field extensions of F to the category Sets of sets. The essential dimension for an arbitrary functor FieldsF → Sets was defined in [5]. One of the applications of the essential dimension is as follows. Suppose we would like to check whether a classification conjecture for the class of objects given by a functor F holds. Usually, a classification conjecture assumes another functor L (a
基本维数和规范维数的纤维维数定理
影响因子：
1.8
作者：
R. Lötscher
通讯作者：
R. Lötscher