Generic Flag Varieties

在代数中，一种非常常见的情况是，一种类型的对象被认为是在某个固定基域的域扩展上。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。它的性质很容易理解，感兴趣的问题也很容易解决。给定类型的一般对象的性质通常太复杂而不能直接处理。理解它的一个重要步骤是研究所谓的类属客体。这可以被认为是最远的相反分裂之一。它的特点是每个对象都是它的特殊化。泛型对象的使用是许多证明中的一个重要组成部分。令人惊讶的是，通用对象比一般对象更容易访问（即，任意的）。同时，类属对象的信息为我们提供了一些关于一般对象的重要信息。泛型对象也为通用架构提供了一个试验场。这个项目建议研究从代数群中产生的一般对象，如下所述。在代数学中，对于某个分裂的半单代数群G，给定类型的对象可以用G-挠子（或主G-齐性空间）来分类，这也是一种常见的情况。在这种情况下，对一般对象的研究就变成了对一般G-torsors的研究。本文的主要目的是对Chow环CH（E/B）有一个完整的认识，其中B是G的Borel子群.商簇E/B是一个光滑的射影簇，称为一般的旗簇。簇E/B的Grothendieck环K（E/B）具有余维支撑滤子.设GK（E/B）是伴随分次环. Chow环CH（E/B）的主要问题是分次环CH（E/B）到GK（E/B）上的标准满同态是否同构。等价地，它询问E/B的联结K理论是否无挠。在这个项目中，我想研究这个问题和一些相关的问题。在这个方向上的突破将是代数群领域的重大成就。这个问题已经得到了回答的积极的几种类型的G。因此，Chow环以及联结K-理论对于这样的G是完全理解的。另一方面，最近已经证明，对于旋量群Spin（17），答案是否定的。其中一个相关的问题是G的分类空间BG的Chow环与BG的K-理论中的表示环R（G）的比较问题。在最近与A. Merkurjev，这个问题已经接近通过引入一个类似的过滤的余维支持R（G）。为了进一步的发展，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.