Roe代数和适用于étale群胚真作用的粗指标映射. I
Roe algebras and coarse index maps for spaces with proper actions of {é}tale groupoids. I
AI总结：
本文为étale群胚真作用空间建立等变Roe代数与K-理论框架，构造粗指标映射的接收函子，并证明满足条件的群胚单纯复形存在通用模。
AI中文摘要：
这是系列论文中的第一篇，旨在将Roe代数和粗指标理论推广到群胚等变情形。我们引入了一些技术，为配备étale群胚真作用的空间建立Roe代数及其K-理论的框架。对于固定的étale群胚G和G-C*-代数A，我们构造了一个函子KC(-; G, A)，从局部紧Hausdorff真G-空间及等变真连续映射的范畴到分次阿贝尔群范畴，该函子为等变粗指标映射提供了一个自然的接收对象。我们研究了通用模的存在性，这是非等变情形中ample模的类比，并发展了一种分解技术来证明它们的存在。作为应用，我们证明了满足适当假设的群胚单纯复形承认通用模。
英文摘要：
This is the first in a series of papers extending Roe algebras and coarse index theory to the groupoid-equivariant setting. We introduce some techniques to develop a framework of Roe algebras and their K-theory for spaces equipped with proper actions of {é}tale groupoids. For a fixed {é}tale groupoid G and G-C* -algebra A, we construct a functor KC(-; G, A) from the category of locally compact Hausdorff proper G-spaces and equivariant proper continuous maps to the category of graded abelian groups, which provides a natural receptacle for an equivariant coarse index map. We study the existence of universal modules, an analog of ample modules in non-equivariant setting, and develop a decomposition technique to establish their existence. As an application, we prove that groupoid simplicial complexes satisfying suitable hypotheses admit universal modules.
