Homology of Steinberg algebras

2024/12/19 by Arnone, Guido, Cortiñas, Guillermo, Mukherjee, Devarshi · 1 citation
#16E40 #19D55 #22A22 #FOS: Mathematics #Group Theory (math.GR) #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2412.15112

Abstract

We study homological invariants of the Steinberg algebra A_k(G) of an ample groupoid G over a commutative ring k. For G principal or Hausdorff with G^\rmIso∖G⁽⁰⁾ discrete, we compute Hochschild and cyclic homology of A_k(G) in terms of groupoid homology. For any ample Hausdorff groupoid G, we find that H_*(G) is a direct summand of HH_*(A_k(G)); using this and the Dennis trace we obtain a map D_*:K_*(A_k(G))→ H_n(G,k). We study this map when G is the (twisted) Exel-Pardo groupoid associated to a self-similar action of a group G on a graph, and compute HH_*(A_k(G)) and H_*(G,k) in terms of the homology of G, and the K-theory of A_k(G) in terms of that of k[G].

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• Equivariant periodic cyclic homology for ample groupoids

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