Noncommutative Projective Algebraic Geometry

摘要/ Abstract本研究处于非交换环理论与代数几何的交叉领域。本课题最抽象的设定是将由一个阿贝尔范畴和其中的一个特殊对象组成的对看作是虚非交换空间上的束的范畴和空间的结构束。这样的一对称为方案。环上的模的种类和环本身就是一个基本的例子。该项目的大部分内容涉及对特殊示例的检查，目的是更好地理解标准几何概念的适当类似物，如子方案、开、闭、奇异等。将特别注意射影空间的非交换类似物；其中包括Sklyanin代数，它最初是由物理学中的一些问题引起的。本文还讨论了非交换环的几个同调性质，如Auslander正则性、Koszul性质和Cohen-Macaulay性质。非交换环理论解决的基本问题是理解非交换变量多项式方程组的（可能是无穷）矩阵的解。非交换代数几何旨在发展一种更几何的方法来解决这样的问题，本质上是通过将解视为几何对象。

Smith, 9701578 Abstract. The proposed research lies at the junction of non-commutative ring theory and algebraic geometry. The most abstract setting for the subject is to view a pair, consisting of an abelian category and a distinguished object in it, as the category of sheaves on some imaginary non-commutative space and the structure sheaf of the space. Such a pair is called a scheme. The category of modules over a ring together with the ring itself is a basic example. Much of the project involves the examination of special examples with the goal of getting a better understanding of the appropriate analogues of standard geometric notions such as subschemes, open, closed, singular, etc. Particular attention will be paid to non-commutative analogues of projective spaces; included in this are the Sklyanin algebras, which first arose from some problems in physics. It is also proposed to examine several homological properties of non-commutative rings, such as Auslander regularity, the Koszul property, and the Cohen-Macaulay property. The basic problem addressed by non-commutative ring theory is to understand the solutions in (possibly infinite) matrices to systems of polynomial equations in non-commuting variables. Non-commutative algebraic geometry aims to develop a more geometric approach to such a problem, essentially by viewing the solutions as geometric objects.