2018/06/09 by Roozbeh Hazrat, Huanhuan Li, Hazrat, Roozbeh +1 · 2 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1806.03398
openalex publication_date 2018/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a graded homology theory for graded étale groupoids. For \mathbb Z-graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients \mathbb Z is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading, our result extends Matui's on zeroth homology of finite graphs with no sinks (shifts of finite type) to all arbitrary graphs. We use our results to show that graded zeroth-homology group is a complete invariant for eventual conjugacy of shift of finite types and could be the unifying invariant for the analytic and the algebraic graph algebras.