Coherent sheaves, sheared D-modules, and Hochschild cochains

TL;DR

This paper establishes a deep connection between ind-coherent sheaves, sheared D-modules, and Hochschild cochains on quasi-smooth schemes and stacks, revealing a Morita equivalence and implications for automorphic functions.

Contribution

It introduces a Morita equivalence between ind-coherent sheaves and sheared D-modules on quasi-smooth schemes and extends this to stacks, linking to automorphic functions.

Findings

Category of ind-coherent sheaves is tensorially related to sheared D-modules.

Proves Morita equivalence between these categories.

Connects the formalism to automorphic functions over function fields.

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.