Derived localisation in algebra and homotopy theory

我们在高中时就开始熟悉本地化(尽管不是用这个名字)。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形成么半群；它是可交换的，因为两个自然数的和不取决于它们相加的顺序。拥有一个既能做加法又能做减法的代数结构是很有用的；这就是整数集Z是如何由自然数构成的；本质上，负数只是简单地加到现有元素上。我们说Z是(仍然可交换的)群，它是么半群N的群完成。还有其他类似的例子：考虑Z的乘法运算(而不是如上所述的加法)。同样，它是交换么半群，它的完备性是有理数的集合Q。请注意，在最后一个例子中，Z支持两种结构：加法和乘法，适当地相容。它的形式化被称为(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看作具有整数分子和分母的分数。么半群的群补全和环的分数域都是局部化的例子。在本项目中，我们集中于环的局部化，或者更一般的称为微分分次环的结构。给定交换环A和A中元素的集合S，可以尝试形式上对其求逆，或定域于S。例如，Z在所有非零整数的集合上的定域化产生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，是证明定理不可缺少的工具。对交换局部化的理解由来已久。相比之下，我们对非对易环(如方阵的环)的局部化的理解则比较零散；各种有用的结构，如形成分数域，要么是不可能的，要么只有在严格的约束下才能成立。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(研究在连续变形下不变的空间的性质)的整个主题是围绕某个范畴(它是非对易代数的推广)的局部化。本课题的主要观点是，一旦非对易环范畴被适当地扩展，交换局部化的形式性质几乎完全被恢复。本课题的目的是利用这一思想的结果，适当地扩展它，并在代数、拓扑和范畴理论中得到结果。

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.