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We consider the crossed product $G_{ga}$ by $\R_+^*$ of the adiabaticgroupoid associated with any Lie groupoid $G$. We construct an explicit Moritaequivalence between the exact sequence of order 0 pseudodifferential operatorson $G$ and (a restriction of) the natural exact sequence associated with$G_{ga}$. As an important intermediate step, we express a pseudodifferentialoperator on $G$ as an integral associated to a smoothing operator on theadiabatic groupoid $G_{ad}$ of $G$.

Adiabatic groupoid, crossed product by $\R_+^*$ and Pseudodifferential calculus