The Brauer-Long group of \({\mathbb{Z}}/p^ t{\mathbb{Z}}\)-dimodule algebras
For a commutative ring R and an abelian group G, \textit{F. W. Long} has defined the notion of a G-Azumaya algebra over R and shown that a smash product of G-Azumaya algebras is again a G-Azumaya algebra [Proc. Lond. Math. Soc., III. Ser. 29, 237-256 (1974; Zbl 0294.13003)]. Equivalence classes of G-Azumaya algebras form a group BD(R,G), now called the Brauer-Long group. Azumaya algebras are G-Azumaya with trivial G-action and trivial G-grading, and the Brauer group B(R) is a subgroup of BD(R,G). Now assume that G is a cyclic group of order \(n=p^ t\) for a prime integer p, \(p\cdot 1\) is a unit in R, and R contains a primitive n-th root of unity if p is odd or a primitive 2n-th root of unity if \(p=2\). Then the equivalence classes of central G-Azumaya algebras form a subgroup BAz(R,G) of index 2 in BD(R,G), and so there is an exact sequence \(BAz(R,G)\to BD(R,G)\to Z/2Z.\) The authors describe a split epimorphism \(\beta\) of BAz(R,G) onto the group Aut(G) of automorphisms of G to obtain an exact sequence \(\ker \beta \to BAz(R,G)\to Aut(G);\) and they derive an exact sequence \(B(R)\to \ker \beta \to Gal(R,GR)\times Gal(R,RG),\) where Gal(R,RG) is the group of isomorphism classes of Galois algebras over R with respect to the group ring RG, i.e. algebras which are Galois with respect to a G-grading, and Gal(R,GR) is the group of isomorphism classes of Galois algebras over R with respect to the dual of the Hopf algebra RG, i.e. algebras which are Galois with respect to an action of G by algebra automorphisms. Therefore, as a set, BD(R,G) is equivalent to the Cartesian product \(Z/2Z\times Aut(G)\times Gal(R,GR)\times Gal(R,RG)\times B(R).\) By analysis of the preceding exact sequences, the authors give a complete description of the multiplication on this Cartesian product corresponding to the multiplication on BD(R,G).
scientific article; zbMATH DE number 4136136
scientific article