ABSTRACT
Let G be an abelian group and consider a commutative ring R graded by G. The so-called arithmetically graded rings focused on in this section will be Krull domains graded by G, but we expound only those properties that are relevant for applications in the study of graded Azumaya algebras over the rings we consider. It will be proved in a subsequent section that for any Azumaya algebra which is G-graded over R, there exists a noetherian subring R’ of R and an Azumaya algebra A’ over R’ such that A = A’ ⊗RR and A’ and R’ are G-graded. Checking the proof of this result, it shows that R’ is actually finitely generated by homogeneous elements over the prime ring of R, hence R’ is graded by some finitely generated subgroup of G. Since A’ is a finite R’-module, it also follows that A’ is graded by some finitely generated subgroup H of G. Since A’ ⊂ A(H) and 20 R’ ⊂ R( H, it follows that A(H) = A’ ⊗R’ R(H) and thus that A(H) is an Azumaya algebra over R(H) , graded by H. This establishes that the study of G-graded Azumaya algebras over R may be reduced to the study of H-graded Azumaya algebras over the ring R(H) , for some finitely generated abelian group H. Since H = Z × Htors , where Htors is finite, it follows that only the cases where G = Z m and G is finite have to be dealt with. Here we specialize to the torsion free case and state some results for torsion free abelian groups, if the restriction to the case G ≅ Z m is not essential.
