On Rham cohomology of locally trivial Lie groupoids over triangulated manifolds

Based in the isomorphism between Lie algebroid cohomology and piecewise smooth cohomology, it is proved that the Rham cohomology of a locally trivial Lie groupoid $G$ on a smooth manifold $M$ is isomorphic to the piecewise Rham cohomology of $G$, in which $G$ and $M$ are manifolds without boundary and $M$ is smoothly triangulated by a finite simplicial complex $K$ such that, for each simplex $\Delta$ of $K$, the inverse images of $\Delta$ by the source and target mappings of $G$ are transverses submanifolds in the ambient space $G$. As a consequence, it is shown that the piecewise de Rham cohomology of $G$ does not depend on the triangulation of the base.