Categorical approach to the Baum-Connes conjecture for étale groupoids

Abstract

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.

Dynamic asymptotic dimension and Matui's HK conjecture

“…With 𝐶(𝜙) as in line (7), we obtain a short exact sequence (12) with morphisms defined by 𝜄 ∶ (𝑎 0 , 𝑏) ↦ (𝑎 0 , 𝑏, 0) and 𝑒 1 ∶ (𝑎 0 , 𝑏, 𝑎 1 ) ↦ 𝑎 1 . Let 𝖼 ∶ 𝐵 → 𝐼𝐵 denote the constant * -homomorphism.…”

“…Let  be the homological ideal (see [38,Definition 2.20 and Remark 2.21]) in 𝐾𝐾 𝐺 defined as the kernel of the restriction functor 𝐹 ∶= Res 𝐺 𝐺 0 ∶ 𝐾𝐾 𝐺 → 𝐾𝐾 𝐺 0 . If we define also 𝐸 ∶= Ind 𝐺 𝐺 0 ∶ 𝐾𝐾 𝐺 0 → 𝐾𝐾 𝐺 (see [7,Section 2.1] for a detailed treatment of this), then (𝐸, 𝐹) form an adjoint pair as established in [6, Section 6] (see also [7,Theorem 2.3]). Define 𝐿 ∶= 𝐸•𝐹 ∶ 𝐾𝐾 𝐺 → 𝐾𝐾 𝐺 , and for an object 𝐵 of 𝐾𝐾 𝐺 , let 𝜖 𝐵 ∈ 𝐾𝐾 𝐺 (𝐿(𝐵), 𝐵) be the counit of adjunction, which is computed explicitly in [7,Theorem 2.3].…”

“…If we define also 𝐸 ∶= Ind 𝐺 𝐺 0 ∶ 𝐾𝐾 𝐺 0 → 𝐾𝐾 𝐺 (see [7,Section 2.1] for a detailed treatment of this), then (𝐸, 𝐹) form an adjoint pair as established in [6, Section 6] (see also [7,Theorem 2.3]). Define 𝐿 ∶= 𝐸•𝐹 ∶ 𝐾𝐾 𝐺 → 𝐾𝐾 𝐺 , and for an object 𝐵 of 𝐾𝐾 𝐺 , let 𝜖 𝐵 ∈ 𝐾𝐾 𝐺 (𝐿(𝐵), 𝐵) be the counit of adjunction, which is computed explicitly in [7,Theorem 2.3]. Now, as a consequence of [44,Proposition 3.1], we see there is an (even) -projective resolution in the sense of [44,Definition 2.14] of the object 𝐴 = 𝐶 0 (𝐺 0 ) in 𝐾𝐾 𝐺 (17) where for each 𝑝 ∈ ℕ, 𝑃 𝑝 ∶= 𝐿 𝑝+1 (𝐴) and…”

“…Now, following [37, Lemma 3.2], the -projective resolution in line ( 17) can be augmented in an essentially unique way to a phantom tower as in line (20). Let 𝐽 ∶ 𝐾𝐾 𝐺 → Ab ℤ∕2 be the functor from 𝐾𝐾 𝐺 to the category of ℤ∕2-graded abelian groups defined by 𝐽 ∶= 𝐾 * •𝑗 𝐺 , where 𝑗 𝐺 ∶ 𝐾𝐾 𝐺 → 𝐾𝐾 is descent (see, e.g., [7,Section 1.3]), and 𝐾 * ∶ 𝐾𝐾 → Ab ℤ∕2 is 𝐾-theory; note that on objects, 𝐽(𝐴) ∶= 𝐾 * (𝐴 ⋊ 𝑟 𝐺). This is a stable homological functor (see [38,Definitions 2.12 and 2.14]), so we may use it to build the ABC spectral sequence of [37,Sections 4 and 5] based on the phantom tower from line (20).…”

“…Unfortunately, this necessitates more notation. Let 𝜄 ∶ 𝐶 0 (𝐺 0 ) → 𝐶 * 𝑟 (𝐺) denote the canonical inclusion, and let 𝐶(𝜄) and 𝐷(𝜄) be the corresponding 𝐶 * -algebras from lines (7) and (11). Define 𝑋 𝐷 to consist of all triples (𝜉 0 , 𝜂, 𝜉 1 ) in 𝐿 2 (𝐺, 𝑠) ⊕ 𝐼 ( 𝐿 2 (𝐺, 𝑠) ⊗ 𝜄 𝐶 * 𝑟 (𝐺)…”