Graded rings and (noncommutative) algebraic geometry

按自然数或整数分次的环是交换和非交换射影几何的基础。利用任意有限生成阿贝尔群的分次环来解决非交换代数几何和堆栈中的一些问题，从而拓宽了这一代数基础。PI将使用由任意有限生成的阿贝尔群分次的交换环作为工具来研究由一般线性群的对角子群构成的全局商的堆栈。环作为商堆的齐次坐标环，使用它作为射影格式的普通齐次坐标环，可以避免堆的一些更技术性的方面。在这种方法适用的堆中有环形堆，特别是加权投影堆，以及一些与弦理论的部分有关的堆。利用这种方法，PI对环堆的Grothendieck群和Picard群有了更简单的证明。在非交换射影代数几何中，PI将使用由有限生成的阿贝尔群分次的环来扩展该主题的范围，提供新的例子，并简化某些方法。堆栈可以被视为温和的非交换空间，并且对于非交换几何来说，将它们视为温和的非交换空间是很重要的。尽管这是Connes的非交换几何程序的重要部分(例如，它没有在非交换代数几何中发挥作用。预计这种代数方法将使一些堆栈更容易为非交换代数几何学者所理解，并表明堆栈与他们自己的关注密切相关。这项拟议的研究建立在PI之前的工作基础上，并与其他非交换代数几何研究人员最近的研究相互作用。物理和数学的一个伟大而宏大的主题是对空间的研究。不是外层空间，而是空间本身，所有活动和不活动都发生在空间中。两千多年来，数学和物理学一直受到这种探索的推动。它似乎没有尽头：技术和数学的进步回答了老问题，但每一个新的有利条件都会引发新的问题。空间总是比想象的更加陌生。我们目前的理解仍然是不够的。一个值得注意的新概念是非对易几何。非对易几何颠倒了代数和几何的通常角色。传统上，一个人有一个几何对象，在空间上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。在这个传统中，环是可交换的：乘积XY等于乘积YX。这是因为x和y的测量值是数字，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不必与yx相同。然后我们说这个代数是非对易的。传统上，n个粒子在3维空间中的位置是由3n个数字编码的。已经提出，可以通过三个n乘n矩阵对该数据进行更好的编码。当三个矩阵相互交换时，它们可以同时对角化，其中的n个对角线元素给出了传统的3n个数，但当三个矩阵不交换时，得到了一个根本不同的非交换代数。然后，我们的目标是理解这个非对易代数告诉我们关于空间的什么。这项研究涉及非交换代数的几何方面，它紧密地模仿了传统的代数几何，这是代数和几何的聚合融合，自古以来一直是数学的中心。

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.