Generic Flag Varieties

在代数中，一种类型的对象是通过一些固定基域的域扩展来考虑的，这是一种非常常见的情况。在给定类型的对象中，通常有一个特别简单的对象——拆分对象。它的性质很好理解，感兴趣的问题很容易解决。给定类型的一般对象的性质通常太复杂，无法直接解决。理解它的一个重要步骤是研究所谓的一般对象。这可以被认为是最相反的分裂。它的特点是每个对象都是它的专门化。一般对象的使用是许多证明的重要组成部分。令人惊讶的是，通用对象比通用（即任意）对象更容易访问。同时，关于泛型对象的信息为我们提供了关于泛型对象的一些重要信息。此外，通用对象为一般猜想提供了一个试验场。这个项目建议研究由代数群产生的一般对象，如下所述。对于某些分裂半单代数群g，给定类型的对象被g -环量（或主g -齐次空间）分类也是代数中常见的情况。在这种情况下，一般对象的研究变成了一般g -环量的研究。本项目的主要目标是实现对Chow环CH（E/B）的完全理解，其中B是g的Borel子群，商品种E/B是一个被称为泛旗品种的光滑投影品种。品种E/B的格罗滕迪克环K（E/B）具有支撑余维的过滤作用。设GK（E/B）为相关联的分级环。关于Chow环CH（E/B）的主要问题是：分级环CH（E/B）在GK（E/B）上的正则满射同构是否为同构。同样地，它询问E/B的连接k理论是否没有扭转。在这个项目中，我想研究这个问题以及一些相关的问题。在这个方向上的突破将是代数群领域的一项重大成就。这个问题对于几种类型的g已经有了正的答案，因此，对于这类g， Chow环和连接k理论都被完全理解了。另一方面，最近有研究表明，对于自旋子群Spin来说，答案是负的（17）。其中一个相关问题是G的分类空间BG的Chow环与作为BG的k理论的表示环R(G)的比较问题。在最近与a . Merkurjev的联合工作中，通过引入R(G)上支持的余维过滤的模拟来解决这个问题。为了进一步的进展，我们计划构建一个g等变版本的连接k理论，特别是BG的连接k理论。

Generic Flag Varieties It is a very common situation in Algebra that a type of objects is considered over field extensions of some fixed base field. Among the objects of the given type, there is usually a particularly simple one -- the split object. Its properties are well-understood and the problems of interest are easily solved for it. The nature of a general object of the given type is usually too complicated to be addressed directly. An important step towards its understanding is the study of the so-called generic object. This can be considered as the farthest opposite to the split one. It is characterized by the property that every object is its specialization. The use of generic objects is an important ingredient in many proofs. Surprisingly, generic objects are more accessible than the general (i.e., arbitrary) ones. At the same time, information on generic objects provides us with some important information on the general objects. Also generic objects provide a testing ground for general conjectures. This project suggests the study of generic objects arising from algebraic groups as described below. It is again a common situation in Algebra that objects of the given type are classified by G-torsors (or principle G-homogeneous spaces), for certain split semisimple algebraic group G. In such situations, the study of generic objects becomes the study of generic G-torsors. The main objective of the present project is to achieve a complete understanding of the Chow ring CH(E/B), where B is a Borel subgroup of G. The quotient variety E/B is a smooth projective variety known as a generic flag variety. The Grothendieck ring K(E/B) of the variety E/B is endowed with the filtration by codimension of support. Let GK(E/B) be the associated graded ring. Main Question on the Chow ring CH(E/B) asks whether the canonical surjective homomorphism of graded rings CH(E/B) onto GK(E/B) is an isomorphism. Equivalently, it asks if the connective K-theory of E/B is free of torsion. In this project, I want to study this question and some related problems. A breakthrough in this direction will be a major achievement in the field of algebraic groups. The question has already been answered by positive for several types of G. As a consequence, the Chow ring as well as the connective K-theory are completely understood for such G. On the other hand, it has been recently shown that the answer is negative for the spinor group Spin(17). One of the related problems is the problem of comparison of the Chow ring of the classifying space BG of G with the representation ring R(G) viewed as the K-theory of BG. In a recent joint work with A. Merkurjev, this problem has been approached by introducing an analogue of the filtration by codimension of support on R(G). For further progress, we plan to construct a G-equivariant version of the connective K-theory and, in particular, the connective K-theory of BG.