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Orientability and complete intersections for rings and ring spectra

环和环谱的定向性和完全交集

基本信息

批准号：
EP/E012957/1
负责人：
John Greenlees
金额：
$ 37.71万
依托单位：
University of Sheffield
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2007
资助国家：
英国
起止时间：
2007 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FE012957%2F1
关键词：
Orientability complete intersections rings ring

项目摘要

Formulating properties and proofs in a robust form has double benefits. Firstly it establishes that they are stable under minor perturbations, and thus record something significant. Secondly, it allows them to be applied more generally.For example, properties of commutative rings are often very rigid and depend on elementwise manipulations. Formulating this in homological terms is a step forward, and then one may hope to state things in the derived category in structural terms. Finally, if this formulation is of an appropriate form, it may also apply to ring spectra. For instance if the property was regularity of a local ring, it is classically formulated in terms of a regular sequence of elements, but characterized by Serre in terms of finite projective dimension, and reformulated in the derived category as being able to build the residue field from the ring in finitely many steps. This then applies to ring spectra, and includes the classifying spaces of p-compact groups in the sense of Dwyer-Wilkerson. Similarly, the notion of Gorenstein commutative rings extends to ring spectra, where it includes Poincar\'e duality for manifolds and Benson-Carlson duality for group cohomology.The project aims to investigate this process further, and in particular for the notion of complete intersection. In fact the properties of rings are often (for example in the above cases) motivated by algebraic geometry. Thus the process described above gives one of the ingredients in a homotopy invariant form of algebraic geometry, now under investigation from various different directions.

以稳健的形式制定属性和证明具有双重好处。首先，它确定了它们在微小扰动下是稳定的，因此记录了一些重要的东西。其次，它使它们的应用更广泛，例如，交换环的性质往往是非常严格的，并依赖于元素操作。用同调术语来表述这一点是向前迈出的一步，然后人们可能希望用结构术语来陈述派生范畴中的事物。最后，如果这个公式是一个适当的形式，它也可以适用于环光谱。例如，如果属性是正则性的一个局部环，它是经典的制定方面的一个经常序列的元素，但其特点塞尔方面的有限投影维数，并重新制定了衍生类别能够建立剩余领域从环在100多个步骤。这也适用于环谱，并且包括Dwyer-Wilkerson意义下p-紧群的分类空间。类似地，Gorenstein交换环的概念扩展到环谱，其中它包括流形的Poincar\'e对偶和群上同调的Benson-Carlson对偶。该项目旨在进一步研究这个过程，特别是完全相交的概念。事实上，环的性质经常（例如在上述情况下）是由代数几何激发的。因此，上述过程给出了代数几何的同伦不变形式中的一个成分，现在正在从各种不同的方向进行研究。