Strongly graded groupoids and strongly graded Steinberg algebras

@article{Clark2017StronglyGG,
  title={Strongly graded groupoids and strongly graded Steinberg algebras},
  author={L. O. Clark and R. Hazrat and S. Rigby},
  journal={arXiv: Rings and Algebras},
  year={2017}
}
We study strongly graded groupoids, which are topological groupoids $\mathcal G$ equipped with a continuous, surjective functor $\kappa: \mathcal G \to \Gamma$, to a discrete group $\Gamma$, such that $\kappa^{-1}(\gamma)\kappa^{-1}(\delta) = \kappa^{-1}(\gamma \delta)$, for all $\gamma, \delta \in \Gamma$. We introduce the category of graded $\mathcal G$-sheaves, and prove an analogue of Dade's Theorem: $\mathcal G$ is strongly graded if and only if every graded $\mathcal G$-sheaf is induced… Expand
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References

SHOWING 1-10 OF 47 REFERENCES
MODULES OVER ÉTALE GROUPOID ALGEBRAS AS SHEAVES
  • B. Steinberg
  • Mathematics
  • Journal of the Australian Mathematical Society
  • 2014
Abstract The author has previously associated to each commutative ring with unit $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \geExpand
The path space of a higher-rank graph
We construct a locally compact Hausdorff topology on the path space of a finitely aligned $k$-graph $\Lambda$. We identify the boundary-path space $\partial\Lambda$ as the spectrum of a commutativeExpand
A Generalised uniqueness theorem and the graded ideal structure of Steinberg algebras
Given an ample, Hausdorff groupoid $\mathcal{G}$, and a unital commutative ring $R$, we consider the Steinberg algebra $A_R(\mathcal {G})$. First we prove a uniqueness theorem for this algebra andExpand
A groupoid generalisation of Leavitt path algebras
Let $$G$$G be a locally compact, Hausdorff, étale groupoid whose unit space is totally disconnected. We show that the collection $$A(G)$$A(G) of locally-constant, compactly supported complex-valuedExpand
CHAIN CONDITIONS ON ÉTALE GROUPOID ALGEBRAS WITH APPLICATIONS TO LEAVITT PATH ALGEBRAS AND INVERSE SEMIGROUP ALGEBRAS
  • B. Steinberg
  • Mathematics
  • Journal of the Australian Mathematical Society
  • 2018
The author has previously associated to each commutative ring with unit $R$ and étale groupoid $\mathscr{G}$ with locally compact, Hausdorff and totally disconnected unit space an $R$ -algebraExpand
A Groupoid Approach to Discrete Inverse Semigroup Algebras
Let $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of functions on the universal \'etale groupoidExpand
Graded Steinberg algebras and their representations
We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorff groupoid. In the first part of the paper, we show that this category is isomorphic to theExpand
Nonstable K-theory for Graph Algebras
We compute the monoid V(LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(E), and we show that this monoid satisfies the refinement property andExpand
Graded Steinberg algebras and partial actions
Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on aExpand
The classifying topos of a continuous groupoid. II
We investigate some properties of the functor B which associates to any continuous groupoid G its classifying topos BG of equivariant G-sheaves. In particular, it will be shown that the category ofExpand
...
1
2
3
4
5
...