  We have investigated the magnetic ordering in the ultrathin c(10$\times$2)
CoO(111) film supported on Ir(100) on the basis of ab-initio calculations. We
find a close relationship between the local structural properties of the oxide
film and the induced magnetic order, leading to alternating ferromagnetically
and anti-ferromagnetically ordered segments. While the local magnetic order is
directly related to the geometric position of the Co atoms, the mismatch
between the CoO film and the Ir substrate leads to a complex long-range order
of the oxide.
