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