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

TL;DR

This paper establishes an equivalence between Lie-Rinehart algebras and certain homotopy classes of negatively graded Lie algebroids, extending the algebraic understanding of singular foliations and their universal structures.

Contribution

It introduces a categorical equivalence linking Lie-Rinehart algebras to acyclic Lie algebroids, broadening the algebraic framework for singular foliations.

Findings

Equivalence of categories between Lie-Rinehart algebras and homotopy classes of Lie algebroids.

Extension of the universal -manifold construction to algebraic settings.

Explicit examples illustrating the theory and its applications.

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.