论文标题
Baum-Connes构想的分类方法典型类固醇
Categorical approach to the Baum-Connes conjecture for étale groupoids
论文作者
论文摘要
我们考虑了与典型类固醇相关的kasparov类别，并且通过利用其三角形结构，我们研究了其在“弱接合”对象上的定位，从而扩展了R. Meyer和R. Nest的先前工作。我们证明了弱缩合对象的子类别是与投影对象的本地化子类别互补的，这些子类别是根据与各向同性相关的某些适当的亚群的“紧凑诱导”代数定义的。由此产生的“强”鲍姆 - 康纳斯猜想意味着经典的猜想，其配方阐明了几种永久性和其他功能语句。我们提出了多种应用，包括对通用系数定理的后果，一个普遍的“降低”原则，对于无限态度可染的类的群体的注射率结果，baum-connes猜想的集体捆绑包，以及关于$ k $ g $ k $ groups twist twisted Groupoid $ c^*$ c^*$ c^*$ algebebras youncom of homeoteper的不变性的结果。
We consider the equivariant Kasparov category associated to an étale groupoid, and by leveraging its triangulated structure we study its localization at the "weakly contractible" objects, extending previous work by R. Meyer and R. Nest. We prove the subcategory of weakly contractible objects is complementary to the localizing subcategory of projective objects, which are defined in terms of "compactly induced" algebras with respect to certain proper subgroupoids related to isotropy. The resulting "strong" Baum-Connes conjecture implies the classical one, and its formulation clarifies several permanence properties and other functorial statements. We present multiple applications, including consequences for the Universal Coefficient Theorem, a generalized "Going-Down" principle, injectivity results for groupoids that are amenable at infinity, the Baum-Connes conjecture for group bundles, and a result about the invariance of $K$-groups of twisted groupoid $C^*$-algebras under homotopy of twists.
