$L_{\mathrm{\infty }}$-resolutions and twisting in the curved context

In this short note we describe an alternative global version of the twisting procedure used by Dolgushev to prove formality theorems. This allows us to describe the maps of Fedosov resolutions, which are key factors of the formality morphisms, in terms of a twist of the fiberwise quasi-isomorphisms induced by the local formality theorems proved by Kontsevich and Shoikhet. The key point consists in considering $L_{\mathrm{\infty }}$-resolutions of the Fedosov resolutions obtained by Dolgushev and an adapted notion of Maurer-Cartan element. This allows us to perform the twisting of the quasi-isomorphism intertwining them in a global manner.