The completion theorem for twisted equivariant K-theory of finite Lie groupoids

Let ${\mathcal G}$ be a finite Lie groupoid, $X$ a ${\mathcal G}$-space, and $P$ a ${\mathcal G}$-stable projective bundle. Write $I_{\mathcal G}$ for the augmentation ideal used in the $I_{\mathcal G}$-adic completion, and let $E{\mathcal G}$ denote the universal ${\mathcal G}$-space. Then there is an isomorphism of $K_{\mathcal G}^*(G_0)$-modules:

$${}^P K_{\mathcal G}^*(X)_{I_{\mathcal G}}^{\wedge}\longrightarrow {}^{P\times_{\pi}E{\mathcal G}}K_{\mathcal G}^*(X\times_{\pi}E{\mathcal G}).$$

The completion theorem. The completed twisted ${\mathcal G}$-equivariant $K$-theory of $X$ is isomorphic to the twisted equivariant $K$-theory after forming the homotopy quotient by $E{\mathcal G}$. This identifies the $I_{\mathcal G}$-adic completion with the corresponding theory on the universal proper ${\mathcal G}$-space; the supplied text does not state a resolution status beyond presenting the result in the completion-theorem section.