Artin主理想环上的各向异性模
Anisotropic modules over artinian principal ideal rings
摘要：
设V是域k上的有限维向量空间，W是1维k-向量空间。设<，>：V × V \to W是对称双线性型.然后<，>被称为各向异性，如果对于所有非零的v \in V，我们有<v，v> \neq 0。受代数数论中一个问题的启发，我们将各向异性的概念推广到了Artin主理想环上的双生成模上的对称双线性型。我们将给出这种各向异性概念的许多等价定义。其中一个定义表明，可以通过检查向量空间上的某些形式是否各向异性来检查形式是否各向异性。我们还将讨论对称双线性型的拟各向异性的概念，它没有有用的向量空间模拟。最后，我们将讨论一个对称双线性型的根，它也没有一个有用的向量空间模拟。所有这三个概念在代数数论中都有应用。
英文摘要：
Let V be a finite-dimensional vector space over a field k and let W be a 1-dimensional k-vector space. Let < , >: V x V \to W be a symmetric bilinear form. Then < , > is called anisotropic if for all nonzero v \in V we have <v,v> \neq 0. Motivated by a problem in algebraic number theory, we come up with a generalization of the concept of anisotropy to symmetric bilinear forms on finitely generated modules over artinian principal ideal rings. We will give many equivalent definitions of this concept of anisotropy. One of the definitions shows that one can check if a form is anisotropic by checking if certain forms on vector spaces are anisotropic. We will also discuss the concept of quasi-anisotropy of a symmetric bilinear form, which has no useful vector space analogue. Finally we will discuss the radical root of a symmetric bilinear form, which doesn't have a useful vector space analogue either. All three concepts have applications in algebraic number theory.
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