Abstract

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 the category of unital
left modules over the Steinberg algebra of the skewproduct groupoid arising from the grading. To do this,
we show that the Steinberg algebra of the skew product is graded isomorphic to a natural generalisation
of the Cohen–Montgomery smash product of the Steinberg algebra of the underlying groupoid with the
grading group. In the second part of the paper, we study the minimal (that is, irreducible) representations in
the category of graded modules of a Steinberg algebra, and establish a connection between the annihilator
ideals of these minimal representations, and effectiveness of the groupoid.
Specialising our results, we produce a representation of the monoid of graded finitely generated
projective modules over a Leavitt path algebra. We deduce that the lattice of orderideals in the K0group
of the Leavitt path algebra is isomorphic to the lattice of graded ideals of the algebra. We also investigate
the graded monoid for Kumjian–Pask algebras of rowfinite kgraphs with no sources. We prove that
these algebras are graded von Neumann regular rings, and record some structural consequences of this.