Higher Grothendieck-Witt groups and A1-homotopy theory

高等 Grothendieck-Witt 群和 A1 同伦理论

基本信息

批准号：
EP/M001113/1
负责人：
Marco Schlichting
金额：
$ 36.71万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2015
资助国家：
英国
起止时间：
2015 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FM001113%2F1
关键词：
Higher Grothendieck Witt groups A1

项目摘要

An inner product space over a commutative ring R is a finitely generated projective R-module equipped with a non-degenerate symmetric bilinear form. Inner product spaces are important everywhere in mathematics but also for instance in physics (e.g., Minkowski space), chemistry (e.g., crystallography) and computer science (e.g., design of codes for a band limited channel).In general, the classification of inner product spaces is a very difficult problem. As an example, the classification of projective modules over the ring of integers Z is easy (there is, up to isomorphism, precisely one for every given rank) whereas the classification of inner product spaces over Z is unknown: for a given rank there are only finitely many isometry classes but we don't know how many (even positive definite) inner product spaces of rank 32 there are over Z.Though still far from being trivial, the study of inner product spaces simplifies when one introduces stable equivalence: two inner product spaces X and Y are stably equivalent if there is a third such space Z and an isometry between the orthogonal sum of X and Z with the orthogonal sum of Y and Z. For instance, two inner product spaces over the ring of integers are stably equivalent if and only if they have the same rank and signature. The set of stable equivalence classes becomes an abelian monoid under orthogonal sum and embeds into the Grothendieck-Witt group GW(R) of formal differences of stable equivalence classes. For many rings (such as fields and local rings in which 2 is a unit) two inner product spaces are isometric if and only if they have the same class in GW(R). For such rings, the classification of inner product spaces thus amounts to computing the group GW(R). The computation of these groups is greatly aided by the fact that they are part of a cohomology theory which allows us to compute GW(R) from "local data".So far, most tools to compute the groups GW(R) only work when 2 is a unit in R which is a (hopefully unnecessary) restrictive assumption. The main objective of the proposal is to develop tools for computing GW(R) that don't need 2 to be a unit in R. A second objective is the study of GW(R) in the context of an algebraic analogue (A1-homotopy theory) of the continuous world around us which was used by Voevodsky in his work on the Bloch-Kato conjecture which won him the Fields medal.

交换环 R 上的内积空间是配备有非简并对称双线性形式的有限生成射影 R 模。内积空间在数学中无处不在，而且在物理学（例如，闵可夫斯基空间）、化学（例如，晶体学）和计算机科学（例如，带限通道的代码设计）中也很重要。一般来说，内积空间的分类产品空间是一个非常困难的问题。举个例子，整数环 Z 上的射影模的分类很容易（直到同构，对于每个给定的等级，精确地有一个），而 Z 上的内积空间的分类是未知的：对于给定的等级，有只有有限多个等距类，但我们不知道 Z 上有多少个（甚至是正定的）32 阶内积空间。虽然还远非微不足道，但当引入稳定等价时，内积空间的研究就变得简单了：两个如果存在第三个这样的空间 Z，并且 X 和 Z 的正交和与 Y 和 Z 的正交和之间存在等距，则内积空间 X 和 Y 稳定等价。例如，整数环上的两个内积空间为当且仅当它们具有相同的等级和签名时才稳定等效。稳定等价类的集合在正交和下成为阿贝尔幺半群，并嵌入到稳定等价类的形式差的 Grothendieck-Witt 群 GW(R) 中。对于许多环（例如域和局部环，其中 2 是一个单位），两个内积空间是等距的当且仅当它们在 GW(R) 中具有相同的类。对于这样的环，内积空间的分类因此相当于计算群 GW(R)。这些群的计算得到了很大的帮助，因为它们是上同调理论的一部分，该理论允许我们从“本地数据”计算 GW(R)。到目前为止，大多数计算群 GW(R) 的工具仅在以下情况下工作2 是 R 中的一个单位，它是一个（希望是不必要的）限制性假设。该提案的主要目标是开发计算 GW(R) 的工具，不需要 2 作为 R 中的一个单位。第二个目标是在代数模拟 (A1- Voevodsky 在他关于布洛赫-加藤猜想的研究中使用了我们周围连续世界的同伦理论，该猜想为他赢得了菲尔兹奖。