Graded rings and (noncommutative) algebraic geometry

由自然数或整数分阶的环是交换和非交换射影代数几何的基础。由其他组评分的环较少使用。PI将通过使用由任意有限生成的阿贝尔群分级的环来解决非交换代数几何和堆栈中的一些问题来扩展这一代数基础。PI将使用由任意有限生成的阿贝尔群分级的交换环作为工具来研究一般线性群的对角子群的全局商堆栈。环充当商堆栈的齐次坐标环，并且像使用射影方案的普通齐次坐标环一样使用它可以避免堆栈的一些更技术性的方面。适用于这种方法的堆栈包括环堆栈，特别是加权投影堆栈，以及一些与弦理论部分相关的堆栈。通过使用这种方法，PI对于环阵的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级的环来扩展该主题的范围，提供新的例子，并简化一些方法。堆栈可以看作是轻度非交换空间，对于非交换几何来说，将它们视为这样是很重要的。虽然这是Connes的非交换几何程序（例如，轨道）的重要组成部分，但它在非交换代数几何中没有发挥作用。预计这种对某些堆栈的代数方法将使非交换代数几何学者更容易理解这一主题，并表明堆栈与其自身关注的问题密切相关。提出的研究建立在PI以前的工作基础上，并与最近在非交换代数几何领域工作的其他人的研究相互作用。对空间的研究是物理学和数学的一个重要主题。不是外层空间，而是空间本身，是所有活动和不活动发生的舞台。两千多年来，数学和物理学一直受到这一探索的推动。似乎没有尽头：技术和数学的进步回答了老问题，但每一个新的有利位置都会引发新的问题。太空总是比想象中更奇怪。我们目前的认识还不充分。一个引人注目的新思想是非交换几何。非交换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，一个空间上的函数环。在传统中，环是可交换的：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不会影响结果——我们说x和y可以互换。但是，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不可交换的。传统上，n个粒子在三维空间中的位置由3n个数字编码。有人提出，用三个n × n矩阵对数据进行更好的编码。当这三个矩阵相互交换时，它们可以同时对角化，其中的n个对角项给出传统的3n个数字。但是当矩阵不能交换时就得到了一些本质上不同的东西，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。提出的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，代数和几何的典范融合，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.