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Coherent sheaves, sheared D-modules, and Hochschild cochains

Dario Beraldo, Kevin Lin, Wyatt Reeves

Abstract

We show that the category of ind-coherent sheaves on a quasi-smooth scheme is naturally tensored over the category of sheared D-modules on its shifted cotangent bundle, commuting with its natural action of categorified Hoschschild cochains. We prove that it defines a Morita equivalence as such. We then extend these results to quasi-smooth Artin stacks. As a consequence of our formalism, we are able to articulate a precise sense in which the space of unramified automorphic functions over a function field localizes over the stack of arithmetic Arthur parameters.

Coherent sheaves, sheared D-modules, and Hochschild cochains

Abstract

We show that the category of ind-coherent sheaves on a quasi-smooth scheme is naturally tensored over the category of sheared D-modules on its shifted cotangent bundle, commuting with its natural action of categorified Hoschschild cochains. We prove that it defines a Morita equivalence as such. We then extend these results to quasi-smooth Artin stacks. As a consequence of our formalism, we are able to articulate a precise sense in which the space of unramified automorphic functions over a function field localizes over the stack of arithmetic Arthur parameters.

Paper Structure

This paper contains 87 sections, 32 theorems, 495 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of Results
  4. Microlocalization
  5. Morita Equivalence
  6. Extension to stacks
  7. Application to Traces
  8. Examples
  9. The case of a shifted vector space
  10. The case of a classifying stack
  11. Miscellania
  12. Conventions
  13. Acknowledgements
  14. Construction of the action
  15. Review of sheaves of categories
  16. ...and 72 more sections

Key Result

Theorem 1.2.3

Theorems & Definitions (32)

  • Theorem 1.2.3: \ref{['thm:morita_D_H']}
  • Proposition 2.3.3
  • Proposition 2.4.1
  • Proposition 3.1.1
  • Proposition 3.1.2
  • Proposition 3.1.3
  • Theorem 3.2.1
  • Corollary 3.2.2
  • Proposition 3.3.1
  • Proposition 3.4.1
  • ...and 22 more