Graded rings and (noncommutative) algebraic geometry

分级环和（非交换）代数几何

• 批准号: 0602347
• 负责人: S. Paul Smith
• 金额: $ 14.4万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0602347&HistoricalAwards=false
• 关键词: Graded; rings; noncommutative; algebraic; geometry

Rings graded by the natural numbers or integers lie at the foundation of commutative and non-commutative projective algebraic geometry. Rings graded by other groups are less frequently used.The PI will broaden this algebraic foundation by using rings graded by an arbitrary finitely generated abelian group to solve some problems in non-commutative algebraic geometry and stacks. The PI will use commutative rings graded by an arbitrary finitely generated abelian group as a tool to study stacks that are global quotients by diagonal subgroups of the general linear group. The ring acts as a homogeneous coordinate ring of the quotient stack and using it as one uses the ordinary homogeneous coordinate ring of a projective scheme allows one to avoid some of the more technical aspects of stacks.Among the stacks amenable to such an approach are toric stacks, especially weighted projective stacks, and some stacks relevant to parts of string theory. By using such an approach the PI has simpler proofs about the Grothendieck group and Picard group for toric stacks. Within non-commutative projective algebraic geometry the PI will use rings graded by a finitely generated abelian group to extend the range of that subject, to provide new examples, and to simplify some of the methods. Stacks can be viewed as mildly non-commutative spaces and it is important for non-commutative geometry to treat them as such.Although this is an important part of Connes's non-commutative geometry program (for example, orbifolds) it has not played a role in non-commutative algebraic geometry.It is anticipated that this algebraic approach to some stacks will make this subject more accessible to non-commutative algebraic geometers and show that stacks are intimately related to their own concerns.The proposed research builds on previous work of the PI and interacts with the recent research of others working in non-commutative algebraic geometry.One of the great and grand themes of physics and mathematics is the study of space.Not outer space, but space itself, the arena in which all activity and inactivity occurs. For over two millennia mathematics and physics have been driven by this quest. There seems no end to it: technological and mathematical advances answer old questions but each new vantage point prompts new questions. Space always proves stranger than imagined.Our present understanding is still inadequate. One remarkable new idea is non-commutative geometry. Noncommutative geometry reverses the usual roles of algebra and geometry. Traditionally one has a geometric object and taking various measurements on the space produces an algebraic object, a ring of functions on the space. In that tradition the ring is commutative: the product xy is equal to the product yx. This is because the measurements x and y are numbers and the order in which multiplies two numbers does not affect the answer---we say that x and y commute. However, if x and y are matrices, not numbers, the order of multiplication matters---xy need not be the same as yx. We then say the algebra is non-commutative. Traditionally the position of n particles in 3-dimensional space is encoded by 3n numbers. It has been proposed that one might better encode that data by three n-by-n matrices. When the three matrices commute with one another they can be simultaneously diagonalized and the n diagonal entries in them give the traditional 3n numbers.But when the matrices do not commute something fundamentally different is obtained, a non-commutative algebra. The goal then is to understand what this non-commutative algebra is telling us about space. The proposed research concerns the geometric aspects of non-commutative algebra.It is closely modeled on traditional algebraic geometry, the paradigmatic blending of algebra and geometry, which has been at the center of mathematics since ancient times.

自然数或整数分次环是交换和非交换射影代数几何的基础。由其他群分次的环很少被使用，PI将通过使用由任意可交换生成的阿贝尔群分次的环来解决非交换代数几何和堆栈中的一些问题，从而拓宽这一代数基础。PI将使用交换环的评分由一个任意生成的阿贝尔群作为工具，研究堆栈的一般线性群的对角子群的全局一致性。 这个环充当商栈的齐次坐标环，并且使用它就像使用投影方案的普通齐次坐标环一样，可以避免栈的一些技术性方面。在适合这种方法的栈中，有环面栈，特别是加权投影栈，以及一些与弦理论相关的栈。利用这种方法，PI对复曲面堆栈的Grothendieck群和Picard群有了简单的证明。在非交换的射影代数几何中，PI将使用由一个非交换生成的阿贝尔群分级的环来扩展这个主题的范围，提供新的例子，并简化一些方法。堆栈可以被看作是轻度非交换空间，这对于非交换几何来说是很重要的，尽管这是Connes的非交换几何程序的重要部分。（例如，orbifolds），它还没有发挥作用，在非交换代数几何。预计，这种代数方法的一些堆栈将使这个问题更容易获得非-交换代数几何学家，并表明堆栈是密切相关的，他们自己的关注。拟议的研究建立在以前的工作PI和互动与最近的研究其他工作在非交换代数几何。物理学和数学的伟大和宏伟的主题之一是研究空间。不是外层空间，而是空间本身，所有活动和不活动发生的竞技场。两千多年来，数学和物理学一直被这种探索所驱动。它似乎没有尽头：技术和数学的进步回答了老问题，但每一个新的Vantage位置都会引发新的问题。空间总是比想象中的要陌生，我们目前的认识还不够。一个引人注目的新思想是非对易几何。非交换几何颠倒了代数和几何的通常角色。传统上，人们有一个几何对象，对空间进行各种测量会产生一个代数对象，即空间上的一个函数环。在那个传统中，环是交换的：乘积xy等于乘积yx。这是因为测量值x和y是数字，两个数字相乘的顺序并不影响答案-我们说x和y是可交换的。然而，如果x和y是矩阵而不是数字，那么乘法的顺序很重要-xy不需要与yx相同。然后我们说这个代数是非交换的。传统上，n个粒子在三维空间中的位置由3 n个数字编码。有人提出，最好用三个n乘n矩阵对数据进行编码。当这三个矩阵彼此可交换时，它们可以同时对角化，其中的n个对角元素给出传统的3 n数，但当矩阵不可交换时，得到了一些根本不同的东西，即一个非交换代数。我们的目标是理解这个非交换代数告诉我们关于空间的什么。该研究涉及非交换代数的几何方面，它紧密地模仿传统的代数几何，代数和几何的范式融合，自古以来一直是数学的中心。