Lie-Rinehart cohomology and integrable connections on modules of rank one

TL;DR

This paper explores the Lie-Rinehart cohomology of rank one modules with connections over algebraic structures, linking it to topological and geometric properties of singularities and providing explicit computations in specific cases.

Contribution

It offers a new interpretation of Lie-Rinehart cohomology in terms of integrable connections and computes this cohomology for certain singularities and hypersurfaces.

Findings

Cohomology relates to topological invariants of singularity links.

Complete computation of cohomology for quasi-homogeneous hypersurfaces.

Interpretation of cohomology in terms of integrable connections.

Abstract

Let $k$ be an algebraically closed field of characteristic 0, let $R$ be a commutative $k$-algebra, and let $M$ be a torsion free $R$-module of rank one with a connection $\nabla$. We consider the Lie-Rinehart cohomology with values in $End_{R}(M)$ with its induced connection, and give an interpretation of this cohomology in terms of the integrable connections on $M$. When $R$ is an isolated singularity of dimension $d\geq2$, we relate the Lie-Rinehart cohomology to the topological cohomology of the link of the singularity, and when $R$ is a quasi-homogenous hypersurface of dimension two, we give a complete computation of the cohomology.