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Derived localisation in algebra and homotopy theory

代数和同伦理论中的导出局域化

基本信息

批准号：
EP/N015452/1
负责人：
Andrey Lazarev
金额：
$ 40.44万
依托单位：
Lancaster University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN015452%2F1
关键词：
Derived localisation algebra homotopy theory

项目摘要

We begin to get acquainted with localisation (albeit not under this name) while in high school. Consider the set N of natural numbers: 0, 1,2,3,... Two natural numbers could be added, but they cannot always be subtracted: 1-2 is not a natural number. We say that N forms a monoid; it is commutative because a sum of two natural numbers does not depend on the order in which they are added. It is useful to have an algebraic structure that accommodates subtraction as well as addition; this is how the set of integers Z is constructed out of natural numbers; essentially, negatives are simply added to the existing elements. We say that Z is a (still commutative) group and it is a group completion of the monoid N.There are other similar examples: consider Z with the operation of multiplication (rather than addition as above). Again, it is a commutative monoid and its completion is the set Q of rational numbers.Note that in the last example Z supports two structures: addition and multiplication, suitably compatible. A formalization of this is called a (commutative) ring. On the other hand, Q is more than a ring: it is a field, which means that all non-zero element of it are invertible. The field Q is called the field of fractions of Z, because its elements could indeed be viewed as fraction with integer numerator and denominator.Group completions of monoids and fields of fractions of rings are examples of localization. In the present project we concentrate on localization of rings, or or more general structures called differential graded rings. Given a commutative ring A and a collection S of elements in A one can try to formally invert it, or localise at S. For example, the localization of Z at the set of all non-zero integers produces Q. This is one of the most fundamental and simple procedures in commutative algebra; it serves as an underpinning of advanced fields of pure mathematics, such as algebraic geometry, and is indispensable as a tool for proving theorems. Commutative localisation has been well understood for a long time.In contrast, our understanding of localization of noncommutative rings (such as a ring of square matrces) is more patchy; various useful constructions, such as forming a field of fractions, are either impossible or only hold under severe constraints. On the other hand, noncommutative localization is very important, for example it could be said that the whole subject of homotopy theory (the study of those properties of spaces which do not change under continuous deformation) revolves around localization of a certain category (which is a generalization of a noncommutative algebra).The main insight of the present project, which builds on a recent work by the proposers, is that once the category of noncommutative rings is suitably extended, the formal properties of commutative localization are almost completely restored.The goal of the present project is to exploit the consequences of this idea, extend it suitably and derive consequences in algebra, topology and category theory.

我们开始熟悉本地化（虽然不是在这个名字下），而在高中。考虑自然数的集合N：0，1，2，3，.两个自然数可以相加，但不能总是相减：1-2不是自然数。我们说N形成幺半群;它是可交换的，因为两个自然数的和不依赖于它们相加的顺序。有一个代数结构，既可以做减法，也可以做加法，这是用自然数构造整数集合Z的方法;本质上，负数只是简单地加到现有的元素上。我们说Z是一个（仍然是交换的）群，并且它是么半群N的一个群完成。还有其他类似的例子：考虑Z的乘法运算（而不是上面的加法运算）。同样，它是一个交换幺半群，它的完备化是有理数的集合Q。注意，在最后一个例子中，Z支持两种结构：加法和乘法，适当地兼容。这种形式化称为（交换）环。另一方面，Q不仅仅是一个环：它是一个域，这意味着它的所有非零元素都是可逆的。域Q被称为Z的分数域，因为它的元素确实可以被看作是具有整数分子和分母的分数。幺半群的群完备化和环的分数域都是局部化的例子。在本项目中，我们集中在局部化的环，或或更一般的结构称为微分分次环。给定一个交换环A和A中元素的集合S，我们可以尝试形式上将它反转，或者局部化在S。例如，Z在所有非零整数的集合上的局部化产生Q。这是交换代数中最基本和最简单的过程之一;它是纯数学高级领域的基础，如代数几何，并且是证明定理不可或缺的工具。交换局部化在很长一段时间内已经得到了很好的理解，相比之下，我们对非交换环（如平方矩阵环）的局部化的理解则比较零散;各种有用的构造，如形成一个分数域，要么是不可能的，要么只有在严格的约束下才成立。另一方面，非对易局部化是非常重要的，例如，可以说，同伦理论的整个主题（研究在连续变形下不发生变化的空间的性质）围绕着某个范畴的局部化（这是一个推广的非交换代数）。本项目的主要见解，它建立在最近的工作，由提案人，本文的目标是利用这一思想的结果，将其适当地推广，并在代数、拓扑和范畴论中得到相应的结果。