Lambda-structures in stable categories

稳定类别中的 Lambda 结构

基本信息

批准号：
EP/I034017/1
负责人：
Vladimir Guletskii
金额：
$ 12.39万
依托单位：
University of Liverpool
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FI034017%2F1
关键词：
Lambda structures stable categories

项目摘要

Motives are objects encoding arithmetic and geometry at the same time. This project is about derived symmetric powers in the Morel-Voevodsky motivic stable category over a field.In mathematics it is important to relate additive and multiplicative operations in an algebraic structure, or in a category of geometrical or arithmetical nature. For example, the Kuenneth rule expresses the multiplicative n-th power of a sum A + B in terms of the sum of products of multiplicative i-th and (n-i)-th powers of A and B respectively. A categorified lambda-structure in a symmetric monoidal triangulated category T is a set of endofunctors of T, indexed by non-negative integers, which behave similarly to a usual algebraic lambda-structure in a commutative ring. In particular, the values of the n-th endofunctor on the vertices in a distinguished triangle are related by means of a tower (called Kuenneth tower) of morphisms in T whose cones can be computed by the Kuenneth rule. Our first aim is to show that if T is an abstract stable homotopy category, i.e. the homotopy category of symmetric spectra over a nice simplicial symmetric monoidal model category C, then left derived symmetric powers do exist in T, and they give a lambda-structure in the above sense, provided some natural symmetrizability assumption on cofibrations in C. Left derived symmetric powers will be homotopical symmetric powers, i.e. homotopy colimits of the action of symmetric groups on monoidal powers. Lambda-structures of left derived symmetric powers bring a powerful computational tool to compute homotopical symmetric powers in many stable homotopy categories. For example, this works well in topology, when T is the homotopy category of the category of topological symmetric spectra.Being applied in the Morel-Voevodsky motivic stable category, such lambda-structures encode deep geometrical and arithmetical properties of algebraic varieties over the ground field, which do not appear in the topological setting. In particular, the relation symmetric powers with the contraction of the affine line to a point, and with operations arising from symmetric powers of algebraic varieties over a field, attract our special attention in this project. Thus, we aim to construct and to study a lambda-structure of left derived symmetric powers in the Morel-Voevodsky motivic stable category, and to use it in order to discover completely new phenomena in arithmetic algebraic geometry and motivic theory.

动机是同时编码算术和几何形状的对象。该项目是关于在字段上的莫雷尔·沃夫斯基动机稳定类别中的派生对称力。例如，Kuenneth规则分别根据a和b的乘法i-th和（n-i）幂的产物总和表示了总和A + B的乘法n-幂。在对称单体三角测量类别中，分类的lambda结构是T的一组t的内膜函数，由非负整数索引，其行为与通常的代数lambda结构相似。特别是，在杰出三角形中的顶点上的n-th-th-th-th-th-thinfunctor的值是通过T塔（称为Kuenneth Tower）的t塔相关的T，其圆锥体可以由Kuenneth Rule计算。我们的第一个目的是表明，如果T是一个抽象的稳定同谱素类别，即对对称光谱的同质拷贝类别C的同型类别C类别C类别C类别，则确实存在T中的左派生对称能力，并且它们在上述意义上给出了lambda结构，在上述意义上提供了一些自然的对符号的象征性，请在C.C.左右的symmetrive symertive symertive c.C.C.C.C.C.C.C.C.C. C. C. C.左侧的c.权力，即对称群体对单型力量的作用的同型。左推导的对称功率的Lambda结构带来了一个强大的计算工具，以计算许多稳定同型类别中的同位对称能力。例如，这在拓扑中效果很好，当T是拓扑对称光谱的类别时，应用于Morel-Voovodsky动机稳定类别中应用的lambda结构编码的深层几何和算术特性编码了代数多品种而不是在地面领域中出现的室外场所，而在室外场景中不像近地设置。特别是，关系对称能力随着仿期线的收缩而达到某个点，以及由代数品种在一个领域的对称能力产生的操作，引起了我们在该项目中的特别关注。因此，我们旨在构建和研究Morel-Voevodsky动机稳定类别中左推导的对称能力的lambda结构，并使用它以发现算术代数的几何学和动机理论中的全新现象。