Formality Theorem with Coefficients in a Module

In this paper, X will denote a manifold. In a very famous paper, Kontsevich Ko] showed that the differential graded Lie algebra (DGLA) of polydifferential operators on X is formal. Calaque C1] extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid E over X, he defined the DGLA of E-polydifferential operators, and showed that it is formal. Denote by the DGLA of E-polyvector fields. Considering M, a module over E, we define the-module of E-polyvector fields with values in M. Similarly, we define the-module of E-polydifferential operators with values in M,. We show that there is a quasi-isomorphism of L _∞-modules over from to . Our result extends Calaque’s (and Kontsevich’s) result.