The purposes of the present invention are: to provide an alumina substrate material that an AlN crystal with a higher quality could be produced by it when aluminum nitride (AlN) crystals or the like are grown on an alumina substrate; to provide an alumina substrate formed with an AlN layer with decreased warping; and to provide a substrate material that, the AlN crystal can be promoted to spontaneously peeled and turn to be independent crystal by it, if excessive stress due to unavoidable lattice mismatching is applied, when the substrate material is used as the seed crystal. By an AlN layer being formed on a surface of the alumina substrate and a rare earth elements-containing layer and/or rare earth elements-containing regions being formed in the interior of the AlN layer or in the interface between the AlN layer and the alumina substrate, the stress on the AlN layer can be alleviated, and the warping thereof can be decreased. Moreover, if AlN crystals are grown by using such a substrate, the grown crystals can be peeled spontaneously to be taken out as an independent crystal.