Abstract
The study of orders over surfaces is an integral aspect of noncommutative algebraic
geometry. Although there is a substantial amount known about orders,
relatively few concrete examples have been constructed explicitly. Of those already
constructed, most are del Pezzo orders, noncommutative analogues of del
Pezzo surfaces, the simplest case. We reintroduce a noncommutative analogue
of the well-known commutative cyclic covering trick and implement it to explicitly
construct a vast collection of numerically Calabi-Yau orders, noncommutative
analogues of surfaces of Kodaira dimension 0. This trick allows us to read
off immediately such interesting geometric properties of the order as ramification
data and a maximal commutative quotient scheme. We construct maximal
orders, noncommutative analogues of normal schemes, on rational surfaces and
ruled surfaces. We also use Ogg-Shafarevich theory to construct Azumaya algebras
and, more generally, maximal orders on elliptically fibred surfaces.