顶点环及其Pierce丛
Vertex rings and their Pierce bundles
摘要：
在第一部分中，我们介绍了顶点环，它与顶点代数（或VOA）的关系与交换结合环与复数上的交换结合代数的关系相同。我们证明了顶点环的特征在于戈达德公理。其中包括VOA理论的协方差公理的推广，该公理涉及自然与任何顶点环相关联的规范的Hasse-Schmidt推导。我们给出了这些公理的几个示例性应用，包括与Virasoro代数相关的顶点环的构造。我们考虑一些类别的顶点环，和顶点环的中心所扮演的角色。在第二部分中，我们将与交换环相关的Pierce丛理论推广到顶点环的情形。这相当于构造了与一个顶点环函联的顶点环的某些约化根丛。我们引入冯诺依曼正则顶点环作为冯诺依曼正则交换环的推广;我们得到了这类顶点环的一个特征，即它们的皮尔斯丛是简单顶点环的丛。
英文摘要：
In part I we introduce vertex rings, which bear the same relation to vertex algebras (or VOAs) as commutative, associative rings do to commutative, associative algebras over the complex numbers. We show that vertex rings are characterized by Goddard axioms. These include a generalization of the translation-covariance axiom of VOA theory that involves a canonical Hasse-Schmidt derivation naturally associated to any vertex ring. We give several illustrative applications of these axioms, including the construction of vertex rings associated with the Virasoro algebra. We consider some categories of vertex rings, and the role played by the center of a vertex ring. In part II we extend the theory of Pierce bundles associated to a commutative ring to the setting of vertex rings. This amounts to the construction of certain reduced etale bundles of vertex rings functorially associated to a vertex ring. We introduce von Neumann regular vertex rings as a generalization of von Neumann regular commutative rings; we obtain a characterization of this class of vertex rings as those whose Pierce bundles are bundles of simple vertex rings.
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