Noncommutative Projective Algebraic Geometry

史密斯，9701578《抽象》。本文的研究处于非对易环论和代数几何的交汇点上。这一主题的最抽象的设定是把由阿贝尔范畴和其中的一个杰出客体组成的一对看作是某个虚构的非对易空间上的层范畴和该空间的结构层。这样的配对被称为阴谋。环上的模范畴以及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例子的研究，目的是更好地理解标准几何概念的适当类似，如子方案、开、闭、奇异等。将特别关注射影空间的非对易类似；其中包括Sklyanin代数，它最初是由一些物理问题引起的。研究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。非对易环理论的基本问题是理解非对易变量的多项式方程组的(可能是无穷的)矩阵的解。非对易代数几何旨在开发一种更几何的方法来解决这类问题，本质上是将解视为几何对象。

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.