A CATEGORICAL APPROACH TO THE BAUM–CONNES CONJECTURE FOR ÉTALE GROUPOIDS

• Published in: Journal of the Institute of Mathematics of Jussieu Vol. 23; no. 5; pp. 2319 - 2364
• Main Authors: Bönicke, Christian, Proietti, Valerio
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.09.2024
• Subjects: Algebra; Approximation; Isotropy; Localization; localization; 46L95; groupoid; 18G80; Going-Down principle; 19K35; triangulated category; Baum-Connes conjecture
• ISSN: 1474-7480; 1475-3030
• DOI: 10.1017/S1474748023000531

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.