Cohomology for small categories: $k$-graphs and groupoids

Abstract

Given a higher-rank graph $\Lambda$, we investigate the relationship between the cohomology of $\Lambda$ and the cohomology of the associated groupoid $G_{\Lambda }$. We define an exact functor between the abelian category of right modules over a higher-rank graph $\Lambda$ and the category of $G_{\Lambda }$-sheaves, where $G_{\Lambda }$ is the path groupoid of $\Lambda$. We use this functor to construct compatible homomorphisms from both the cohomology of $\Lambda$ with coefficients in a right $\Lambda$-module, and the continuous cocycle cohomology of $G_{\Lambda }$ with values in the corresponding $G_{\Lambda }$-sheaf, into the sheaf cohomology of $G_{\Lambda }$.

A flaw in the proof of Proposition 4.2 in v1 of this paper has invalidated Proposition 4.8 and Theorem 4.9 from v1. This version (v3) has been substantially revised and includes new results. Version 4 to appear in Banach J. Math. Anal

References in corpus (7)

• Higher rank graph C*-algebras
• Simplicity of C*-algebras associated to higher-rank graphs
• Periodicity in Rank 2 Graph Algebras
• Removing sources from higher-rank graphs
• Rank-two graphs whose C^*-algebras are direct limits of circle algebras
• Entropy of shifts on higher-rank graph C*-algebras
• Graph-Based Models for Kirchberg Algebras