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 [25,27]. 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. Contents 38 3.3. Universal Lie ∞-algebroids associated to an affine variety W 40 References 41

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This paper shows that a weak symmetry action of a Lie algebra g on a singular foliation F induces a unique (up to homotopy) Lie ∞-morphism from g to the DGLA of vector fields on a universal N Q-manifold over F. We deduce from this general result several geometric consequences. For instance, we give an example of a weak action of Lie algebra, i.e., an action on the leaf space which cannot be turned into an action on the ambient space. Last, we introduce the notion of tower of bi-submersions over a singular foliations and lift symmetries to those.