Classically the Chern-classes of a locally free coherent A-module W are defined using the curvature of a connection. If we more generally consider the problem of defining Chern-classes where W is a coherent A-module, a connection might not exist. In this paper we use the linear Lie-algebroid of W, where W is any coherent A-module of finite presentation, to define the first Chern-class of W. We also do explicit calculations of Chern-classes for maximal Cohen-Macaulay modules on isolated hypersurface-singularities and 2-dimensional quotient-singularities.
Classical Chern-Weil theory proves existence of a ring homomorphism - the Chern character - from the Grothendieck ring of locally free finite rank A-modules to the algebraic de Rham cohomology of A. In this paper we prove existence of a ring homomorphism from the Grothendieck ring of locally free finite rank A-modules with an L-connection to the Lie-Rinehart cohomology of L, where L is a Lie-Rinehart algebra. As a corollary we get the classical Chern character.
We study sheaves of Lie-Rinehart algebras over locally ringed spaces. We introduce morphisms and comorphisms of such sheaves and prove factorization theorems for each kind of morphism. Using this notion of morphism, we obtain (higher) homotopy groups and groupoids for such objects which directly generalize the homotopy groups and Weinstein groupoids of Lie algebroids. We consider, the special case of sheaves of Lie-Rinehart algebras over smooth manifolds. We show that, under some reasonable assumptions, such sheaves induce a partition of the underlying manifold into leaves and that these leave…
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 algeb…
In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. We used this notion to define the universal enveloping algebra of the category of $\tilde{L}$-connections and to define the cohomology and homology of an arbitrary connection. In this note we introduce the canonical quotient $L$ of a D-Lie algebra $\tilde{L}$ and use this to classify D-Lie algebras where $L$ is projective as $A$-module. We define for any 2-co…
In this paper we generalize classical results on Lie algebras and universal enveloping algebras of Lie algebras to Lie-Rinehart algebras. We define for any Lie-Rinehart algebra $L$ and any cocycle $f$ in $Z^2(L,B)$, a universal enveloping algebra $U(B,L,f)$ with the property that the category of left modules on $U(B,L,f)$ is equivalent to the category of modules with an $L$-connection where the curvature has "type $f$". A connection of curvature "type $f$" is a special case of a non-flat connection. We also study deformations of filtered algebras and prove a relationship between the deformatio…
The aim of this note is to prove various general properties of a generalization of the full module of first order differential operators on a commutative ring - a $\operatorname{D}$-Lie algebra. A $\operatorname{D}$-Lie algebra $\tilde{L}$ is a Lie-Rinehart algebra over $A/k$ equipped with an $A\otimes_k A$-module structure that is compatible with the Lie-structure. It may be viewed as a simultaneous generalization of a Lie-Rinehart algebra and an Atiyah algebra with additional structure. Given a $\operatorname{D}$-Lie algebra $\tilde{L}$ and an arbitrary connection $(ρ, E)$ we construct the u…
