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Cohomology of Lie algebroids over algebraic spaces

Abhishek Sarkar

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 Čech 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.

Cohomology of Lie algebroids over algebraic spaces

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 Čech 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.

Paper Structure

This paper contains 15 sections, 17 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie algebroids over algebraic spaces
  4. Universal enveloping algebroid of a Lie algebroid
  5. $\mathcal{O}_X/{\mathbb{K}_X}$-bialgebras
  6. Lie algebroid Cohomology over $a$-spaces
  7. Atiyah algebroid
  8. Chevalley-Eilenberg-de Rham complex and Koszul-Rinehart resolution
  9. Chern classes:
  10. Logarithmic de Rham cohomology
  11. Hochschild Cohomology for a Lie algebroid over $a$-spaces
  12. Hochschild hypercohomology of an $\mathcal{O}_X/{\mathbb{K}_X}$-bialgebra
  13. Hochschild hypercohomology of universal enveloping algebroid
  14. Hochschild hypercohomology of jet algebroid
  15. Remarks on Lie algebroids over Schemes.

Key Result

Theorem 3.6

Let $\mathcal{L}$ be a locally free Lie algebroid over $(X, \mathcal{O}_X)$ and $\mathcal{E}$ a representation of $\mathcal{L}$. Then we get an isomorphism of graded vector spaces

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Remark 2.8
  • Remark 2.12
  • Definition 2.13
  • Remark 3.3
  • ...and 31 more