On the exactness of groupoid crossed products

Let \(({\mathcal {A}},G,\alpha )\) be a separable groupoid \(C^*\)-dynamical system and \(C_0(G^{(0)},{\mathcal {A}})\) the \(C^*\)-algebra of continuous sections that vanish at infinity. When \(({\mathcal {A}},G,\alpha )\) has the approximation property, we prove that the crossed product \({\mathcal {A}}\rtimes _{\alpha ,r}G\) is exact if and only if \(C_0(G^{(0)},{\mathcal {A}})\) is exact. In particular, if G is topologically amenable and \(C_0(G^{(0)},{\mathcal {A}})\) is exact, then \({\mathcal {A}}\rtimes _{\alpha ,r}G\) is exact.