Homology of Steinberg algebras
Guido Arnone, Guillermo Corti\~nas, Devarshi Mukherjee

TL;DR
This paper investigates the homological invariants of Steinberg algebras associated with ample groupoids, providing explicit computations of Hochschild and cyclic homology in various cases and relating them to groupoid and group homology.
Contribution
It offers new formulas for Hochschild and cyclic homology of Steinberg algebras, linking them to groupoid homology and extending to self-similar group actions.
Findings
Hochschild and cyclic homology computed for principal or Hausdorff groupoids.
Groupoid homology appears as a direct summand of Hochschild homology.
Established a connection between K-theory of the algebra and groupoid homology.
Abstract
We study homological invariants of the Steinberg algebra of an ample groupoid over a commutative ring . For principal or Hausdorff with discrete, we compute Hochschild and cyclic homology of in terms of groupoid homology. For any ample Hausdorff groupoid , we find that is a direct summand of ; using this and the Dennis trace we obtain a map . We study this map when is the (twisted) Exel-Pardo groupoid associated to a self-similar action of a group on a graph, and compute and in terms of the homology of , and the -theory of…
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Taxonomy
TopicsAdvanced Topics in Algebra · Advanced Operator Algebra Research · Algebraic structures and combinatorial models
