Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.29879

Roe algebras and coarse index maps for spaces with proper actions of {é}tale groupoids. II

Kai Mao
View original source ↗Read source document ↗Export metadata ↓

Abstract

Building on the Roe algebra framework developed in the first paper of this series, we construct equivariant coarse index maps for locally compact Hausdorff spaces equipped with proper actions of {é}tale groupoids, with coefficients in any separable groupoid C* -algebra. The construction uses the coarse structures determined by the groupoid action and requires no choice of metric. By identifying Roe algebras of canonical modules with reduced crossed products, we prove that the Baum-Connes assembly map with coefficients factors through the coarse index map.

Explore related subjects

math.KTmath.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kai Mao. 2026-10-21. Roe algebras and coarse index maps for spaces with proper actions of {é}tale groupoids. II. https://arxiv.org/abs/2610.29879

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The universal property of graded $KK^G$-theory

To each Kasparov cycle $z$ we attach inclusions $B\xrightarrow{j_z}Y_z\xrightarrow{i_z}X_z$ satisfying $[z]j_zi_z=s_zi_z$. A lift of a second factor along $k_z=j_zi_z$ therefore reduces the product to precomposition by a homomorphism. An additional even symmetry gives a retraction of $k_z$; this includes all cycles between trivially graded algebras and the splitting cycles of graded split extensions. The resulting cycle presentation recovers the ungraded equivariant Kasparov category. For arbitrary gradings, explicit tensor and Dirac relations give fusion by passing to a shifted and doubled cycle. We provide concrete Dirac cycles and a compact verification of the two scalar product identities used in this argument. Finally, an index obstruction shows that the corner of the original cycle need not admit a retraction, even after replacing that cycle by an equivalent representative.

math.KT↗
arXiv2026

Additive invariants of Finite-dimensional DG-algebras

Orlov has shown that additive invariants such as K-theory and Hochschild homology of smooth finite-dimensional DG-algebras behave similarly to those of finite-dimensional algebras. The similarities only go so far: there are examples of finite-dimensional DG-algebras with vanishing and uncountable Grothendieck groups. We show that for DG-quivers with relations, the Grothendieck groups are free of the expected rank and that, in the smooth case, all additive invariants are determined by the base field. We use our results to rule out certain finite-dimensional DG-algebras from being phantoms.

math.KT↗
arXiv2026

An explicit Dennis trace and its compatibility with Koszul duality

In this paper, we construct explicit Dennis trace maps from algebraic $K$-theory to Hochschild homology for dg module and curved dg comodule categories, and prove their compatibility with derived Koszul duality. For this, we construct a natural chain map $τ$ from the chains on the newly defined thickened nerve to the total complex of the dg cyclic nerve, with explicit formulas incorporating morphisms and coherent higher homotopies, which generalizes McCarthy's approach. This $τ$ construction works both for modules over dg algebras and for cdg comodules over counital conilpotent curved dg coalgebras. Combining the $τ$ with the dg-enriched Waldhausen $S_\bullet$-construction and a Hochschild delooping theorem yields the Dennis trace maps. For thick subcategories corresponding under derived Koszul duality, we prove that the induced $K$-theory equivalences and Hochschild homology isomorphisms fit into natural commutative diagrams with these trace maps.

math.KT↗