A cohomological interpretation of the graded Brauer group. II (Q1065079)

[For part I see Commun. Algebra 11, 2129-2148 (1983; Zbl 0532.13004).] Using a graded-equivalence relation the set of \({\mathbb{Z}}\)-graded Azumaya algebras over a \({\mathbb{Z}}\)-graded ring may be made into a group with respect to the (graded) tensor product; in this way one obtains the so- called graded Brauer group introduced in 1979 by the reviewer. This group is essentially different from the ''graded'' Brauer group in the sense of C.T.C. Wall (a.o.) where finite grading groups are considered (and where the algebras considered need not be common Azumaya algebras). It is well- known that the Brauer group Br(R) of a commutative ring R embeds in the ''étale Brauer group'' \(H^ 2(R,U)\). The main problem studied in this paper is whether \(Br^ g(R)\) embeds in an étale cohomology group \(\lim_{\to}H^ 2(S/R,U_ 0)\) where the limit is over graded étale coverings of R and \(U_ 0\) is the multiplicative group of units of degree zero of R. In this paper the author avoids the generalization of a theorem of M. Artin concerning joins of Henselian rings by restricting to quasistrongly graded rings, an ugly name for a rather nice and broad class of graded rings. In fact under different conditions one may obtain two different embeddings in étale cohomology groups (a graded one and another one in degree zero as mentioned above but under somewhat more restrictive conditions), a situation that reflects the fact that there exist two versions of a graded crossed product theorem (contained in part I of this paper, loc. cit.). Using a description of the Grothendieck group of the category of graded progenerators and some sequences of Mayer-Vietoris type the author establishes a version of Gabber's theorem \((Br^ gR=H^ 2_ g(R,U)_ 0)_{tors})\) for quasistrongly graded rings using a modification of the version of the proof of Gabber's theorem given by \textit{M. A. Knus} and \textit{M. Ojanguren} [in Groupe de Brauer, Sémin., Les Plans-sur-Bex 1980, Lect. Notes Math. 844, 210-228 (1981; Zbl 0457.13003)]. In more recent work (a monograph which is to appear) the theory of the graded Brauer groups has been extended further by the author and the reviewer and some aspects of the theory in the paper under review have been put in a somewhat different framework (e.g. Verdier coverings etc...)