Lie-Rinehart algebra $\simeq$ acyclic Lie $\infty$-algebroid

Abstract

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.

arXiv comment41 pages. References added, typos corrected, and several sections improved following the referee's remarks. Accepted in "Journal of Algebra"

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