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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上的内积空间是一个具有非退化对称双线性型的生成投射R-模。内积空间在数学中到处都很重要，但在物理学中也很重要（例如，Minkowski空间）、化学（例如，晶体学）和计算机科学（例如，通常，内积空间的分类是一个非常困难的问题。作为例子，整数环Z上投射模的分类是容易的（直到同构，对于每个给定的秩都有一个），而Z上的内积空间的分类是未知的：对于一个给定的秩，只有1000个等距类，但我们不知道有多少（甚至是正定的）秩为32的内积空间在Z上存在。虽然还远不是平凡的，但是当引入稳定等价时，内积空间的研究简化了：两个内积空间X和Y是稳定等价的，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的正交和之间是等距的。例如，整数环上的两个内积空间是稳定等价的当且仅当它们具有相同的秩和签名。稳定等价类的集合在正交和下成为一个交换幺半群，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是单位），两个内积空间是等距的当且仅当它们在GW（R）中有相同的类。对于这样的环，内积空间的分类就相当于计算群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，允许我们从“局部数据”计算GW（R）。到目前为止，大多数计算群GW（R）的工具只在2是R中的一个单位时才起作用，这是一个（希望是不必要的）限制性假设。该提案的主要目标是开发用于计算GW（R）的工具，这些工具不需要2作为R中的单位。第二个目标是研究GW（R）的背景下，代数模拟（A1-同伦理论）的连续世界在我们周围使用Voevodsky在他的工作布洛赫-加藤猜想这为他赢得了菲尔兹奖。