Graded Brauer groups of a groupoid with involution

摘要

We define a group BrR containing, in a sense, the graded complex and orthogonal Brauer groups of a locally compact groupoid equipped with an involution. When the involution is trivial, we show that the new group naturally provides a generalisation of Donovan-Karoubi's graded orthogonal Brauer group GBrO. More generally, it is shown to be a direct summand of the well-known graded complex Brauer group. In addition, we prove that BrR identifies with a direct sum of a Real cohomology group and the abelian group ExtR(, S~1) of Real graded S~1-central extensions. A cohomological picture is then given.