Using the methods of atomic force and electron microscopy and the magneto-optical Kerr effect, the role of the interface, roughness, and thickness of the magnetic layer in the temperature-dependent magnetic properties of thin Al2O3–Co films with a naturally oxidized cobalt surface was studied. The layers were deposited by magnetron sputtering. The thickness of the cobalt layer varied from 2 to 100 nm. For the first time, the dependences of coercive forces and exchange displacements on the thickness of the cobalt film in the temperature range from 80 to 300 K were obtained and analyzed. The contribution to the coercive force and exchange displacement from the oxidized cobalt surface increases as the temperature decreases below 160 K. The magnitude of the contribution depends on the base material on which the cobalt film is deposited and is maximum for a cobalt film with a thickness of ∼20 nm in the Al2O3/Co structure. A weakly magnetic layer was found at the Al2O3/Co interface. The behavior of the exchange bias in this layer is similar to the behavior of a ferromagnetic Co core with a naturally oxidized CoO shell. The thickness of this layer depends on the speed and order of deposition of the layers. When the order of deposition of layers (Co/Al2O3) changes, the behavior of the exchange displacement of the interface becomes similar to that observed in the ferromagnet/antiferromagnet system. That is, when the deposition order changes, the value of the exchange shift changes sign when the cobalt layer thickness is below 10 nm.