  We show that there is an equivalence of categories between Lie-Rinehart
algebras over a commutative algebra $\mathcal O $ and homotopy equivalence
classes of negatively graded Lie $\infty $-algebroids over their resolutions
(=acyclic Lie $\infty$-algebroids). This extends to a purely algebraic setting
the construction of the universal $Q$-manifold of a locally real analytic
singular foliation of Lavau-C.L.-Strobl. In particular, it makes sense for the
universal Lie $\infty$-algebroid of every singular foliation, without any
additional assumption, and for Androulidakis-Zambon singular Lie algebroids.
Also, to any ideal $\mathcal I \subset \mathcal O $ preserved by the anchor map
of a Lie-Rinehart algebra $\mathcal A $, we associate a homotopy equivalence
class of negatively graded Lie $\infty $-algebroids over a complex computing
${\mathrm{Tor}}_{\mathcal O}(\mathcal A, \mathcal O/\mathcal I) $. Several
explicit examples are given.
