Groupoid algebras as Cuntz-Pimsner algebras

We show that if G is a second countable locally compact Hausdorff étale groupoid carrying a suitable cocycle c:G → ℤ, then the reduced C*-algebra of G can be realised naturally as the Cuntz-Pimsner algebra of a correspondence over the reduced C*-algebra of the kernel G₀ of c. If the full and reduced C*-algebras of G₀ coincide, we deduce that the full and reduced C*-algebras of G coincide. We obtain a six-term exact sequence describing the K-theory of C*_r (G) in terms of that of C*_r (G₀).