Cohomology of Lie algebroids over algebraic spaces

TL;DR

This paper develops a cohomological framework for Lie algebroids over algebraic spaces, expressing hypercohomology as a derived functor and exploring Hochschild hypercohomology for related structures.

Contribution

It introduces a derived functor approach to hypercohomology of Lie algebroids and defines Hochschild hypercohomology for sheaves of generalized bialgebras, extending classical theorems.

Findings

Hypercohomology expressed as a derived functor.

Simplification of hypercohomology via Čech cohomology.

Hochschild-Kostant-Rosenberg theorem for Lie algebroids.

Abstract

We consider Lie algebroids over an algebraic space (or topological ringed space) as quasicoherent sheaves of Lie-Rinehart algebras. We express hypercohomology for a locally free Lie algebroid (not necessarily of finite rank) as a derived functor, and simplify it via \v{C}ech cohomology. Furthermore, we define the Hochschild hypercohomology of a sheaf of generalized bialgebras and study the cases of the universal enveloping algebroid and the jet algebroid of a Lie algebroid. In the sequel, we present a version of Hochschild-Kostant-Rosenberg theorem for a locally free Lie algebroid, as well as its dual version.