2021/06/25 by Laurent-Gengoux, Camille, Louis, Ruben
#18G10 #53C12 #53D17 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2106.13458
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 ∞ -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 Lavau-C.L.-Strobl. 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 \mathcal I ⊂ \mathcal O preserved by the anchor map of a Lie-Rinehart algebra \mathcal A , we associate a homotopy equivalence class of negatively graded Lie ∞ -algebroids over a complex computing Tor\mathcal O(\mathcal A, \mathcal O/\mathcal I) . Several explicit examples are given.