Groupoids and Noncommutative Geometry Seminar 

06/06/2007, 09:30 — 16:00 — Room P3.10, Mathematics Building
Alex Kumjian, University of Nevada, Reno

The Brauer group of a locally compact groupoid

We define the Brauer group \(\operatorname{Br}(G)\) of a locally compact groupoid \($G\) to be the set of Morita equivalence classes of pairs \((\mathcal{A},\alpha)\)consisting of an elementary C*-bundle \(\mathcal{A}\) over \(G^{(0)}\) satisfying Fell\'s condition and an action \(\alpha\) of \(G\) on \(\mathcal{A}\) by \(\ast\)-isomorphisms. When \(G\) is a transformation groupoid, then \(\operatorname{Br}(G)\) is the equivariant Brauer group of [1].

It is shown that \(\operatorname{Br}(G)\) is isomorphic to \(\operatorname{Ext}(G,\boldsymbol{T})\), as defined by Renault. It follows that if \(G\) and \(H\) are equivalent groupoids, then \(\operatorname{Br}(G)\) and \(\operatorname{Br}(H)\) are isomorphic.

If \(G\) is étale, then \(\operatorname{Br}(G) \cong H^2(G, \boldsymbol{T})\), where \(H^\ast(G, \cdot)\) denotes the the natural extension of Grothendieck's equivariant sheaf cohomology to étale groupoids. The assignment of such a cohomology class to a pair \((\mathcal{A},\alpha)\) may be viewed as a generalized Dixmier--Douady invariant.

1. A. Kumjian, P. Muhly, J. Renault, D. Williams. The Brauer group of a locally compact groupoid, Amer. J. Math. 120 (1998) 901—954.