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A universal sheaf of algebras governing representations of vector fields on quasi-projective varieties

Yuly Billig, Colin Ingalls

TL;DR

The paper addresses the challenge of generalizing D-modules to capture representations of the Lie algebra of vector fields on smooth quasi-projective varieties. It introduces AV-modules and constructs a universal sheaf of algebras by forming AV = A \# U(V) and then completing to a topologically quasi-coherent object AV̂ via jets of vector fields, Ĵ, in étale charts. The main results establish local structure theorems: Ĵ ≅ V ⋉ (A \widehat{\otimes} \mathcal{L}_+) and AV̂ ≅ D(A) ⊗_A U_A(A \widehat{\otimes} \mathcal{L}_+), along with a localization formula and a finite-level filtration yielding AV^m. This framework unifies AV-modules with D-module techniques, providing a universal, coordinate-independent approach to study representations of vector fields on quasi-projective varieties and enabling a natural passage from local to global theory via topological coherence and jet methods.

Abstract

We construct a quasi-coherent sheaf of associative algebras which controls a category of $AV$-modules over a smooth quasi-projective variety. We establish a local structure theorem, proving that in étale charts these associative algebras decompose into a tensor product of the algebra of differential operators and the universal enveloping algebra of the Lie algebra of power series vector fields vanishing at the origin.

A universal sheaf of algebras governing representations of vector fields on quasi-projective varieties

TL;DR

The paper addresses the challenge of generalizing D-modules to capture representations of the Lie algebra of vector fields on smooth quasi-projective varieties. It introduces AV-modules and constructs a universal sheaf of algebras by forming AV = A \# U(V) and then completing to a topologically quasi-coherent object AV̂ via jets of vector fields, Ĵ, in étale charts. The main results establish local structure theorems: Ĵ ≅ V ⋉ (A \widehat{\otimes} \mathcal{L}_+) and AV̂ ≅ D(A) ⊗_A U_A(A \widehat{\otimes} \mathcal{L}_+), along with a localization formula and a finite-level filtration yielding AV^m. This framework unifies AV-modules with D-module techniques, providing a universal, coordinate-independent approach to study representations of vector fields on quasi-projective varieties and enabling a natural passage from local to global theory via topological coherence and jet methods.

Abstract

We construct a quasi-coherent sheaf of associative algebras which controls a category of -modules over a smooth quasi-projective variety. We establish a local structure theorem, proving that in étale charts these associative algebras decompose into a tensor product of the algebra of differential operators and the universal enveloping algebra of the Lie algebra of power series vector fields vanishing at the origin.
Paper Structure (5 sections, 20 theorems, 75 equations)

This paper contains 5 sections, 20 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Étale charts
  3. Rinehart enveloping algebras
  4. Jets
  5. Completion of ${\mathcal{AV}}$

Key Result

Theorem 1

Let $U \subset X$ be an étale chart. Then Here ${\mathcal{L}}_+$ is the Lie algebra of vector fields on ${\mathbb A}^N$ vanishing at the origin and $A(U) \widehat{\otimes} {\mathcal{L}}_+$ is a completion of the tensor product.

Theorems & Definitions (40)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Lemma 4
  • Definition 5
  • Lemma 6
  • Lemma 7
  • proof
  • Definition 8
  • Theorem 9
  • ...and 30 more