Maximal graded orders over crystalline graded rings

Crystalline graded rings are generalizations of certain classes of rings like generalized twisted group rings, generalized Weyl algebras, and generalized skew crossed products. Under certain conditions, in particular, the group is finite, it is proven that the global dimension of a crystalline graded ring coincides with the global dimension of its base ring. When, in addition, the base ring is a commutative Dedekind domain, two constructions are given for producing maximal graded orders. On the way, a new concept appears, so-called, spectrally twisted group. Some general properties of it are studied. At the end of the paper several examples are considered.