Lie-Rinehart algebras ≃ acyclic Lie ∞-algebroids

Abstract

We show that there is an equivalence of categories between Lie-Rinehart algebras over a commutative algebra O and homotopy equivalence classes of negatively graded Lie ∞-algebroids over their resolutions (=acyclic Lie ∞-algebroids). This extends to a purely algebraic setting the construction of the universal Q-manifold of a locally real analytic singular foliation of [26,28]. In particular, it makes sense for the universal Lie ∞-algebroid of every singular foliation, without any additional assumption, and for Androulidakis-Zambon singular Lie algebroids. Also, to any ideal I⊂O preserved by the anchor map of a Lie-Rinehart algebra A, we associate a homotopy equivalence class of negatively graded Lie ∞-algebroids over complexes computing TorO(A,O/I). Several explicit examples are given.