Tracing projective modules over noncommutative orbifolds

For an action of a finite cyclic group $F$ on $n$-dimensional noncommutative torus $A\_\theta$, we give sufficient conditions when the fundamental projective modules over which determine range canonical trace extend to crossed product $C^\*$-algebra $A\_\theta \rtimes F$. Our results allow us understand F$, and it completely for several examples including products 2-dimensional tori with groups flip $\mathbb Z\_2$ any torus. As application, simple Morita equivalence class \mathbb Z\_2,$ in terms $A\_\theta.