An etale groupoid's invariant measures pair canonically with its zeroth Crainic-Moerdijk homology, yielding a groupoid Elliott invariant shown to match the C*-algebraic Elliott invariant for many integer actions and orbit-breaking constructions.
Groupoid homology and K-theory for algebraic actions from number theory
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abstract
We compute the groupoid homology for the ample groupoids associated with algebraic actions from rings of algebraic integers and integral dynamics. We derive results for the homology of the topological full groups associated with rings of algebraic integers, and we use our groupoid homology calculation to compute the K-theory for ring C*-algebras of rings of algebraic integers, recovering the results of Cuntz and Li and of Li and L\"uck without using Cuntz-Li duality. Moreover, we compute the K-theory for C*-algebras attached to integral dynamics, resolving the conjecture by Barlak, Omland, and Stammeier in full generality.
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A trace pairing and Elliott invariant for groupoid homology
An etale groupoid's invariant measures pair canonically with its zeroth Crainic-Moerdijk homology, yielding a groupoid Elliott invariant shown to match the C*-algebraic Elliott invariant for many integer actions and orbit-breaking constructions.