Noncommutative Projective Algebraic Geometry

史密斯，9701578 抽象的。 该研究处于非交换环理论和代数几何的结合点。对这个主题来说，最抽象的设定是把一个由阿贝尔范畴和其中的一个特殊对象组成的对看作是某个想象的非对易空间上的层范畴和这个空间的结构层。这样的一对称为方案。 环上的模范畴和环本身是一个基本的例子。 该项目的大部分内容涉及特殊的例子检查，目的是更好地理解标准几何概念的适当类似物，如子方案，开放，封闭，奇异等，特别注意将支付给射影空间的非交换类似物;包括在这是Sklyanin代数，它首先出现在物理学中的一些问题。它还提出了检查几个非交换环的同调性质，如Auslander正则性，Koszul性质和Cohen-Macaulay性质。 非交换环理论的基本问题是理解非交换变量的多项式方程组在（可能是无限的）矩阵中的解。 非交换代数几何的目的是发展一个更几何的方法来解决这样的问题，基本上是通过查看的解决方案作为几何对象。

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.