Universal D-modules and stacks of \'etale germs of n-dimensional varieties

TL;DR

This paper develops a framework for classifying étale germs of n-dimensional varieties using stacks, establishing their relation to universal D- and O-modules, and explores their automorphism groups and representation theory.

Contribution

It introduces stacks for étale germs of pointed n-dimensional varieties and links quasi-coherent sheaves on these stacks to universal D- and O-modules, extending Artin's approximation theorem.

Findings

Stacks classify étale germs of n-dimensional varieties.

Quasi-coherent sheaves correspond to universal D- and O-modules.

Automorphism groups of formal discs are central to the structure.

Abstract

We introduce stacks classifying \'etale germs of pointed n-dimensional varieties. We show that quasi-coherent sheaves on these stacks are universal D- and O-modules. We state and prove a relative version of Artin's approximation theorem, and as a consequence identify our stacks with classifying stacks of automorphism groups of the n-dimensional formal disc. We introduce the notion of convergent universal modules, and study them in terms of these stacks and the representation theory of the automorphism groups.